Thermal Shock Induced Fracture
in Ceramic Materials
Saurabh Sudhir Kulkarni
CE22MTECH14009
M.Tech Thesis Stage-1 evaluation
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Contents
• Introduction
• Literature Review
• Motivation
• Objectives
• Formulation
• Results
• Tasks Ahead
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Introduction
Ceramics
• ‘Keramikos’  ‘for pottery
• Inorganic, metallic oxides, carbides or nitride materials.
• High hardness, high temperature resistance, high wear
resistance, high corrosion resistance and lower density but
also brittleness.
• Applications in aerospace and defence industry at extreme
conditions such as thermal shock.
Thermal shock
• Rapid change in temperature that results in very high heat
fluxes and transient mechanical load on the structure.
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Introduction
• Investigation of thermal-shock induced fracture behaviour of
ceramics through experiments , analytical models and numerical
simulations
• Critical thermal stress theory based on thermoelasticity
• Thermal shock damage theory based on fracture mechanics
• Predicting critical thermal shock temperature difference
• Experimental Multi crack propagation with periodic and
hierarchical characteristics in brittle and thin strips through
simulating the thermal shock by water quenching
• Numerical Behaviour of ceramics subjected to thermal shock
through damage mechanics model in the frame of finite element
method, peridynamic model and phase field methods
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Ref. Simulation of crack patterns in quasi-brittle
materials under thermal shock using
4 phase field
and cohesive zone models
Introduction
• Various methods : Cohesive zone modelling, Extended finite element method, Meshfree methods,
Peridynamics, Cracking particle method, Screened poisson equation, Scaled boundary finite element
method, Generalised finite element method and Phase field method.
• Phase field method is used in this work as it is superior to other methods in following aspects
1.
It can naturally handle complex crack patterns and phenomena like crack branching and coalescence
without the need to explicitly track the crack surface.
2.
The method can be used for intrinsic fracture nucleation and propagation.
3.
This method can also be coupled with other methods such as cohesive zone modelling and extended
FEM.
4.
The method can be extended to 3D problems without major difficulty
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1. Experiments and numerical simulations of thermal shock crack patterns in thin
circular ceramic specimens
Authors – Liu Y., Wu X., Guo Q., Jiang C., et al. Year - (2015)
• Observation of periodical cracks development through experimental study on water quenching of
thin circular alumina specimens
• Observation that the higher the temperature, the more the cracks.
• The long cracks become longer, and the short cracks become shorter as initial temperature
increases.
• The two stability criteria of crack propagation, i.e. the minimum potential energy principle and
the fracture mechanics bifurcation theory are in good agreement with the experimental findings.
Crack pattern observed
at temperature of 4000c
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Crack pattern observed
at temperature of 5000c
Temperature vs Spacing
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2. Numerical model for the cracking behavior of heterogeneous brittle solids
subjected to thermal shock
Authors - Tanga S.B., Zhanga H., Tanga C.A., H.Y. Liub H.Y. Year - 2016
• Considered the effect of heterogeneity in the material strength by randomly
specifying the strength and elastic modulus to the elements which follow Weibull
distribution which in turn is obtained by Monte-Carlo simulations.
• The greater conductivity results in smaller thermal gradient and significantly
reduces the number of cracks initiated on the surface.
• The results show that the temperature fluctuates sharply on both sides of the
cracks and is significantly different from the temperature field in the intact
specimen.
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Results showing disturbed temperature field
after evolution of crack
Heterogeneity of Elastic
Modulus assigned over
the domain
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3. Fracture of thermo-elastic solids: Phase-field modeling and new results with an
efficient monolithic solver
Authors – Mandal T. K., Nguyen P. B., Wu J. Y. et al. Year - 2021
• Phase-field regularized cohesive zone model (PF-CZM) for thermo-elastic solids has been
presented which can model thermally induced fracture for both brittle and quasi brittle
formulations.
• BFGS (Broyden–Fletcher–Goldfarb–Shanno) scheme to is used solve multi-field discretized
scheme for efficiency.
• However, the degradation in conductivity with evolution of damage and convective heat transfer
is not accounted for in this work.
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Crack pattern
observed at
temperature of 3000c
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4. Three-dimensional phase-field modeling of temperature-dependent thermal shockinduced fracture in ceramic materials
Authors -Dingyu Li, Peidong Li, Weidong Li, Weiguo Li, Kun Zhou Year -2022
• Thermo-mechanical fracture of alumina specimen subjected to thermal shock is simulated
• Inclusion of temperature dependent fracture energy threshold and consideration of water entry
posture improved the results significantly.
• Crack propagation in 3d specimen also was simulated.
• Simulation results revealed the mechanism that the tensile part of the strain energy controlled
the initiation and propagation of the thermal shock-induced cracks in ceramics.
Crack pattern
observed at
temperature of 3000C
Crack pattern observed
at temperature of
5000C
Three-Dimensional Model
at temperature of 3400C
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Motivation
• Complicated operational environments such as aerodynamic heating on ceramics
can result in tremendous temperature gradient in an instant.
• The inherent brittleness of ceramics makes them vulnerable to thermal shock
induced fracture which can lead to sudden catastrophic failure of structure.
• The analysis of thermal shock induced fracture in ceramics can provide insights
into their failure mechanism.
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Ceramic radomes are used to protect
missile antenna
Alumina samples after water quenching
test
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Motivation
• A comprehensive approach considering all relevant factors has been lacking.
• Focus on individual aspects such as
1. Water quenching
2. temperature-dependent failure criteria
3. Heterogeneity in material properties
Novelty
An improved regularization of the crack to accurately model the phenomenon.
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Objectives
1.To develop a numerical model to simulate thermal shock induced fracture in
ceramic materials.
2. Perform parametric studies on the model by varying parameters such as
• Geometry
• Magnitude of loading
• Material properties
3. Validation of the model with results from the literature
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Formulation
• Consider an infinite one-dimensional (1D) bar with a cross section Γ
and a fully opened crack at x = 0
Sharp Crack
• The sharp crack is regularised using
d(x) = e−|x|/lc
-------- 1
lc = length scale parameter
• The width of the diffuse crack region will increase with lc
• sharp crack will be recovered as lc
0.
Diffused Crack
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Formulation
• The approximation is solution to equation
d(x) − lc2Δd(x) = 0
Dirichlet boundary condition: d(0) = 1 and d(±∞) = 0.
• The Galerkin type weak form
1
2
I(d) = ∫Ω( d2 + lcd’2 )dV.
•
dV = Γdx , I(d = e−|x|/lc) = lcΓ
• Fracture surface density per unit volume is given by
Γ(d) =
1
1
I(d)= ∫Ω(
𝑙𝑐
2𝑙𝑐
d2 + lcd’2 )dV=∫Ω γ( d, d′ )dV
where γ(d, d′ ) is the 1D crack surface density function
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Formulation
• γ(d, ∇d) =
1 2
d
2𝑙𝑐
+ lc|∇d|2
--------------2
• Brittle fracture of thermo-elastic solids is governed by minimizing the free energy L that contains the
elastic energy, fracture energy, thermal energy, heat, and external work
L = − ∫Ω ψε(εe (u, T))dV − ∫ Γc GcdΓ + ∫Ω( ρCpT˙ + ∇⋅Q − q )dV + ∫ ∂Ωt t⋅udΓ + ∫Ω b⋅udV + ∫∂ΩQQ’⋅ndΓ --------3
Where Q is the heat flux
Q’ is denoted as the heat flux at the boundary ∂ΩQ
T is the temperature
The boundaries for heat transfer and mechanical deformation problems are expressed by
∂Ω =∂ΩQ∪∂ΩT and ∂Ω = ∂Ωt ∪ ∂Ωu
t represents the traction vector applied at ∂Ωt
q is the internal heat source
The inertia is ignored because of its imperceptible effect in thermal shock-induced fracture problems.
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Formulation
Gc is the critical energy release rate
ρ is the density
Cp is the specific heat of the material
The term ψε(εe(u, T)) is the elastic energy density relied on the linear strain tensor εe(u, T) = ε − εT
in which the total strain tensor is ε=(∇u +∇uT)/2
and the thermal strain tensor is εT = αΔTI,
where α is thermal expansion coefficient of materials,
ΔT is the temperature difference and I is the identity tensor.
By introducing crack density function in fracture energy
G
∫ΓcGcdΓ = ∫Ω c (d2 + lcd’2 )dV.
2𝑙𝑐
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------------4
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Formulation
• As a crack does not propagate under compression, the strain energy density that drives the crack
propagation should be decomposed into tensile and compressive parts.
• A degradation function g(d) is introduced into the positive part of the strain energy density to
characterize the effect of the phase-field crack on material properties.
• Therefore, the strain energy density has the form ψε(εe (u, T), d ) = g(d)ψ+ε(εe ) + ψ−ε (εe )
Where g(d) =(1−d)4 + δ
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Formulation
Strong Form
• divσ + ρb = 0 in Ω,
• ρCpT˙ + ∇ ⋅ Q = q in Ω,
• Gclc(d−lc2Δd ) = g’ (d) ψ+ε in Ω,
• σ ⋅ n = t on ∂Ωt,
• Q ⋅ n = Q on ∂ΩQ ,
• ∇d ⋅ n = 0 on ∂Γc.
With initial conditions
T(x, t) = T0(x, 0) in Ω,
u(x, t) = u0(x, 0) in Ω,
d(x, t) = d0(x, 0) in Ω
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----------- 5a
----------- 5b
------------ 5c
------------ 6a
------------ 6b
------------- 6c
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Formulation
• According to Eq. 6c, the phase-field variable d will evolve if ψe+> 0,
even if ψe+ has a very small value
• Therefore, fracture energy is modified as
∫ΓcGcdΓ =∫Ω[1 − g(d)]ψc + ψc( d2 + lc2|∇d|2 )dV
Where ψc is critical fracture energy threshold
Temperature dependent fracture energy threshold can be given as
∫0TCp(T)dT
ψc(T) = ψc (T0) [ 1 − Tm
]
∫0 Cp(T)dT
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Formulation
Weak Form
• ∫Ω ρCpT˙δTdV − ∫ΩQ ⋅ ∇(δT)dV − ∫Ω qδTdV + ∫∂ΩQ QδTdΓ = 0
• ∫Ω σ : δεdV + ∫∂Ωt t ⋅ δudΓ + ∫Ω b ⋅ δudV = 0,
• ∫Ω g’ (d)HeδddV + ∫Ω2ψc ( dδd − lc2∇d∇(δd) )dV = 0
where He(u(x,s), T) = max s∈[0,t] 〈 ψe+(u(x,s),T) − ψc 〉
δT, δu and δd are the variational test functions.
Due to weak coupling effects and high calculation efficiency the equations are solved in staggered
manner .
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Formulation
Approximation with FEM shape functions
T = ∑nI=1 NTi Ti
u = ∑nI=1 Nui ui
d = ∑nI=1 Ndi di
Gradients of the field variables can be given by using geometry matrices as
∇T = ∑nI=1 BTi Ti
ε = ∑nI=1 Bui ui
∇d = ∑n I=1 Bdi di
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Formulation
Residuals can be obtained as
RTi = ∫Ω ρCpT˙NTI dV − ∫ΩQ BTI dV − ∫Ω qNTI dV + ∫∂ΩQ QNTI dA
Rui = ∫ΩσBuIdV - ∫∂Ωt tNuIdΓ - ∫ΩbNuIdV
Rdi = ∫Ω2ψclc2BdI∇ddV + ∫Ω2NdI [ ψcd − 2(1 − d)3He ]dV
Corresponding tangent matrices can be obtained as
∂RTi
KTij=
= ∫Ω ρCpΔt( NTi )TNTj dV + ∫Ω (1 − d)4 k( BTi )TBTjdV
∂T𝑗
∂R
Kuij = ui = ∫Ω[(1 − d)4 + δ ]( Bui )TD0BujdV
∂𝑢𝑗
∂R
Kdij = di = ∫Ω 2ψclc2( Bdi )T BdjdV + ∫Ω2( Ndi )TNdj[ ψc + 6(1 − d)2He ] dV
∂𝑑𝑗
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Problem Statement
• A 1mm*1mm thin plate with unit thickness subjected to Dirichlet Boundary condition of T=288K
on all the boundaries following initial conditions.
T=288 K
T(x, 0) = 773K in Ω
u(x, 0) = 0 in Ω
d(x, 0) = 0 in Ω.
T=288 K
T0=773K
uo=0
do=o
T=288 K
T=288 K
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Input parameters used in analysis
• Density = 3.95 g/cc;
• Heat capacity = 764e-3;
• Thermal conductivity = 36;
• Heat transfer coefficient = 0.04;
• Coefficient of thermal expansion =5e-6;
• Young’s Modulus = 378000 N/mm2;
• Poisson’s ratio = 0.22;
• Critical Fracture Energy Threshold = 0.1397;
• Element type – Linear Quadrilateral;
• Element size – 0.01mm;
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Mesh used in the analysis
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Results
Temperature distribution when Dirichlet Boundary
condition of 288k and initial condition of 773k was
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applied.
When initial cracks were introduced at
midpoints of boundaries and Dirichlet
boundary condition of 600 k was applied
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Tasks Ahead
• Improve the accuracy of model by introducing convection at the boundaries.
• Improve the accuracy by changing the regularisation of the crack.
• Introduce heterogeneity for better accuracy.
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Thank You
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