refractories WORLDFORUM 4 (2012) [1] 1 1 Introduction Refractories are generally subjected to com- bined thermomechanical loading. The chal- lenging aim in this research field is to devel- op refractory structures with materials prop- erties matched for specific applications. Above all, the thermal shock resistance is the one mechanical property, which has to be improved. This requires an understanding Furthermore, stress intensity factors (SIF) are by calculated using the submodel technique and alternatively the energy release rate is calculated from a conservation line-integral. Results are presented in terms of numerical simulations of damage patterns at different conditions. 2 Theoretical framework We consider a solid continuum with thermo- mechanical initial and boundary conditions given by stresses t → and heat flux q → (Neu- mann) or displacement u → and temperature θ (Dirichlet). To incorporate local microstruc- tural features, mesoscale cell models with linear elastic matrix properties are intro- duced generally containing voids, cracks or grain boundaries. The cell model with boundary ∂V describes a Representative Volume Element (RVE) [1, 2] in the continu- um. In the homogenization process general- ly we obtain effective elastic (C * ijkl ) and ther- mal properties (λ * ij and α * ij ) of an RVE. It is essential that the size of the RVE is chosen according to the condition L >> d >> b . For brittle materials typical values of an RVE of the influence of the microstructure. With- in the framework of continuum mechanics, it is possible to develop models at the macro level of the material and structural behav- iour by introducing effective tensors, which may contain a detailed representation of the microstructure and account for thermo- mechanical equilibrium on the microlevel. In connection with numerical methods, stress, deformation and damage at thermo- mechanical loading can be determined for arbitrary structures and boundary value problems. However, little work has been done in this field with respect to refractory materials. The aim of this study is to present a simple microcrack based damage model for brittle materials under thermo-mechanical dynam- ical loading conditions. To combine fracture- and damage-mechanical approaches, sub- models containing a sharp crack tip are introduced in the FEM model at the ends of the damage zones. Within the submodels numerical and analytical approaches can be integrated, representing interactions be- tween macro-cracks and microstructure. Multiscale Modeling for the Simulation of Damage Processes at Refractory Materials under Thermal Shock D. Henneberg, A. Ricoeur D. Henneberg, A. Ricoeur University of Kassel Institute of Mechanics 34125 Kassel Germany Corresponding-Author: D. Henneberg E-mail: [email protected]Keywords: damage mechanics, multiscale modelling, micro cracks Received: 06.10.2011 Accepted: 24.10.2011 A brittle material damage model based on the theoretical concept of continuum damage mechanics is presented. Cell models are developed including microcrack initiation and growth. To combine fracture- and damage-mechanical ap- proaches, submodels containing a sharp crack tip are introduced at the ends of the damage zones. Also, a conservation inte- gral is applied yielding the en- ergy release rate of an equiva- lent macro-crack. Fig. 1 Problem formulation on different scales and concept of homogenization (C * ijkl : effective elastic tensor, λ * ij : effective thermal conductivity, α * ij : effective coefficients of expansion, q ⎯ → : heat flux tensor, t ⎯ → : traction tensor) Refractories worldforum; ISSN 1868-2405; 2012, Volume 4, Issue 1, S. 1-4
Transcript
refractories WORLDFORUM 4 (2012) [1] 1
1 Introduction
Refractories are generally subjected to com-bined thermomechanical loading. The chal-lenging aim in this research field is to devel-op refractory structures with materials prop-erties matched for specific applications.Above all, the thermal shock resistance isthe one mechanical property, which has tobe improved. This requires an understanding
Furthermore, stress intensity factors (SIF) areby calculated using the submodel techniqueand alternatively the energy release rate iscalculated from a conservation line-integral.Results are presented in terms of numericalsimulations of damage patterns at differentconditions.
2 Theoretical framework
We consider a solid continuum with thermo-mechanical initial and boundary conditionsgiven by stresses t
→ and heat flux q→ (Neu-mann) or displacement u→ and temperature θ(Dirichlet). To incorporate local microstruc-tural features, mesoscale cell models withlinear elastic matrix properties are intro-duced generally containing voids, cracks orgrain boundaries. The cell model withboundary ∂V describes a RepresentativeVolume Element (RVE) [1, 2] in the continu-um. In the homogenization process general-ly we obtain effective elastic (C*
ijk l ) and ther-mal properties (λ*
ij and α*ij ) of an RVE. It is
essential that the size of the RVE is chosenaccording to the condition L >> d >> b .For brittle materials typical values of an RVE
of the influence of the microstructure. With-in the framework of continuum mechanics, itis possible to develop models at the macrolevel of the material and structural behav-iour by introducing effective tensors, whichmay contain a detailed representation of themicrostructure and account for thermo-mechanical equilibrium on the microlevel. Inconnection with numerical methods, stress,deformation and damage at thermo-mechanical loading can be determined forarbitrary structures and boundary valueproblems. However, little work has beendone in this field with respect to refractorymaterials.The aim of this study is to present a simplemicrocrack based damage model for brittlematerials under thermo-mechanical dynam-ical loading conditions. To combine fracture-and damage-mechanical approaches, sub-models containing a sharp crack tip are introduced in the FEM model at the ends of the damage zones. Within the submodelsnumerical and analytical approaches can beintegrated, representing interactions be-tween macro-cracks and microstructure.
Multiscale Modeling for the Simulation of Damage Processes at Refractory Materials under Thermal Shock
A brittle material damagemodel based on the theoreticalconcept of continuum damagemechanics is presented. Cellmodels are developed includingmicrocrack initiation andgrowth. To combine fracture-and damage-mechanical ap-proaches, submodels containinga sharp crack tip are introducedat the ends of the damagezones. Also, a conservation inte-gral is applied yielding the en-ergy release rate of an equiva-lent macro-crack.
Fig. 1 Problem formulation on different scales and concept of homogenization (C*ijkl: effective elastic tensor, λ*ij: effective thermal conductivity, α*ij: effective coefficients of expansion, q⎯ →: heat flux tensor, t⎯ →: traction tensor)
are 0,1 mm, which is related to the mi-crostructural size scale of real material [3].The principle procedure of a homogenizationprocess at refractory materials under thermalshock is illustrated in Fig. 1.To derive average stress and strain tensorsfor the inhomogeneous field, we considertwo subdomains with different properties,i.e. the defect or crack phase with volumeVc, interface S and unit normal nj inside thematrix material with volume Vm and surface∂V, see Fig. 2. The faces of the infinitely thinmicrocrack are separated according to thepositive (x 2 > 0) and negative (x 2 < 0) halfspaces as S = S+ + S–. The vector of thedisplacement jump is defined as:
(1)
The basic equation for an average macro-scopic stress field in a simply connected domain V is given as:
(2)
where x j,k = δjk and σik ,k = 0 in the case ofquasi static crack growth and without the
action of body forces. Applying Gauss’s theorem, stress is transformed to the sur-face. We get the average stress ⟨σij⟩ with ti
as the traction vector and ∂V as the bound-ary of the RVE:
(3)
For the average stress in both subdomainsVc and Vm according to Fig. 2 the previousequation leads to:
(4)
The t mi and t c
i describe the tractions at theboundaries of matrix and defect volumes.Due to continuity of tractions at the interfacethe last term in equation (4) is disappearing. The equation of volume average macro-strain can according be given as:
(5)
Applying the superposition principle andconsidering linear elastic behavior for thematerial matrix, the average strain is decom-posed into a part due to the matrix and onedue to the defect phase. In case of micro-cracks of zero stiffness inside the materialmatrix, equation (5) leads to the result:
(6)
with ⟨εij⟩M as average strain in the sur-rounding matrix of volume Vm = cmV andthe displacement jump at the crack interfaceS– = S+ according to equation (1). For a de-fect phase consisting of microcracks, the fac-tor cM = 1. The last term of equation (6) ❬εij❭C describes the average strain of the de-fect phase. Neglecting microcrack interaction, we con-sider a single crack with initial length 2a inan infinite domain under mixed-mode load-ing, see Fig. 3.Mixed-mode loading is considered due to ar-bitrary crack orientations. However, thosecracks are assumed to be most critical, withrespect to growth whose faces are perpen-dicular to the maximum principle stress. Theaverage strain of a microcrack defect phase
The strain in the x 2 – direction is obtainedfrom equation (7) as:
(8)
and the shear deformation as:
(9)
with [4] asdisplacement jump for Mode-I and Mode-IIloading and Ε as Young’s modulus of thematrix material. In the following, the ratio 4a2
—A = f will be introduced as damage variable or crack density parameter. If f = 1, the microcrackspans the whole RVE, thus the material is lo-cally damaged. The macroscopic averagestrain of the crack phase according to theequations (8) to (9) becomes:
(10)
Equation (10) represents an anisotropic ma-terial law for the defect phase. The effectiveinelastic material law for the damaged ma-terial evolves from equation (6) and leads toa generalized Hook’s law:
(11)
Here, (C*ijk l )-1 denotes the effective compli-
ance tensor. The criterion for microcrack evolution hasbeen chosen in equivalence to a classical R-curve based Mode-I macro crack growthcriterion [5]:
(12)
with KI as Mode-I SIF depending on localstress and the crack length a and KR de-pending on the crack propagation length a.Considering e.g. a damage zone at the tip ofa macroscopic crack (Fig. 4) we have twopossible states. If the damage variable fholds the initial value f0, the material isisotropic assuming a statistical distributionof orientations of microcracks. If the damagevariable is increasing (f > f0) the material
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Fig. 3 A single crack in an infinite domainunder mixed-mode loading, A is the area of the RVE
refractories WORLDFORUM 4 (2012) [1] 3
becomes anisotropic due to cracks orienta-ted perpendicularly to the direction of prin-ciple tensile stress σI growing faster thanothers. Those are considered relevant andthus dominate in the model for damage re-gions, see Fig. 4. Therefore, a transformationof the effective elastic tensor C*
ijk l with re-spect to the local crack coordinate system(x 1, x 2) needs to be done. In our continuumdamage model, there aren’t any macro-scopic cracks in terms of free surfaces as de-picted in Fig. 4, in fact the crack itself con-sists of a slender damage zone.For the isotropic case, microcracks areopened in all directions within the x 1 – x 2
plane, thus the applied stresses are:
(13)
They depend on the initial damage variablef0 and the macroscopic strain εij. For theanisotropic case the matrix of macro stress isthe following:
(14)
The essential thermal parameters of refrac-tory materials which influence reliability andlife time are thermal conductivity λij, ther-mal expansion αij and specific heat cH. Inthis work, hysteresis loops have been mod-eled for α(θ), λ(θ) and cH(θ) covering atempera-ture range from 20 °C to 1200 °C.Thermally induced stresses are calculatedfrom Hooke’s law introducing the tempera-ture change Δθ:
(15)
where ε totij is the total strain and εel
k l denotesthe elastic strain. In order to simulate ther-mal stress, the temperature distribution inthe material is required. Therefore, the ther-mal field problem:
(16)
is solved first, supplying a transient tempera-ture field as loading quantity for the mech-anical boundary value problem.To describe interactions between a damagezone representing a macroscopic crack and
the microstructure it is possible to apply asubmodel technique. Since a tip of a damagezone does not exhibit a singularity, it is feas-ible to include a sharp crack tip in a sub-model, which is introduced in the globalmodel at the end of the damage zone. Thus,fracture- and damage-mechanical approach-es are combined in only one numerical FEM-simulation (Fig. 5). In general, there is a clos-ure effect due to a finite stiffness at the in-tegration points belong to the damage zone,which can be illustrated as spring elementsbetween crack faces. This effect can be ob-served by experimental analysis of thermalshock. Another method to determine the stress in-tensity factors is to calculate the J-integral. Acommercial implementation cannot be ap-plied here, since there are no real crack facesin terms of free surfaces. Thus, a line integralis calculated with an integration contourreaching from one boundary of the damagezone to the other. For a Mode-I loading with
the direction of crack extension zk , the SIFevolves from [5]:
(17)
Crack surface integrals cannot be intro-duced, so results of equation (17) are pathdependent.
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Fig. 4 Schematic representation of a damage zone with equally distributedparallel cracks inside the RVE and localand global coordinate systems
Fig. 5 Global model under tensile load (left) and damage zone with crack tip submodeland displacement boundary condition ui from global model (right)
Fig. 6 Damage zone at Mode-I loading (σ22 = 96 MPa) with crack tip submodel and integration contours
4 refractories WORLDFORUM 4 (2012) [1]
3 Numerical examples
The anisotropic, non-linear material law isimplemented into the Finite Element CodeABAQUS using a Subroutine UMAT. Fig. 6shows the FE-model of a plate with tensileloading σ22. The damage zone initiating atthe notch is growing like a macroscopiccrack of length a. At the tip of the damagezone the submodel and two integrations
contours for the J-integral are presented. InFig. 7 KI is calculated for different cracklengths applying the submodel techniqueand the J-integral. Comparing the results,both methods yield similar values. Here, itshould be taken into account that bothmethods cannot be exact transferring frac-ture mechanical concepts to continuumdamage mechanics. The blue line represents
the handbook solution for an ideal crack.Whereas bridging is taken into account atthe damage model via finite stiffness of thedamage zone, the handbook solution isbased on traction-free crack surfaces, thusleading to much higher values of KI.As a second test geometry, we take a platewith temperature jump Δθ = 1500 K at thetop surface, Fig. 8. Of course, thermal shocksimulations have to account for temperature-dependent material data and inertia effects.In a simulation with constant param-eterswe observe equally spaced crack nucleationstarting close to the surface. In the simula-tion with temperature dependent par-ame-ters λ(θ), cH(θ) and α(θ) the crack initiationstarts underneath the surface. In any casedamage zones are initiated at locations withhighest temperature and stress gradients.
4 Summary
A continuum damage model for refractoriesceramics is presented incorporating fracturemechanical approaches. Results shows dam-age patterns under thermal shock condi-tions.
Acknowledgment
Financial support by the German ScienceFoundation (DFG) within the SPP 1418 isgratefully acknowledged.
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