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Open Access 2024 | OriginalPaper | Buchkapitel

23. Determination of the Temperature-Dependent Fracture and Damage Properties of Ceramic Filter Materials from Small Scale Specimens

verfasst von : Martin Abendroth, Shahin Takht Firouzeh, Meinhard Kuna, Bjoern Kiefer

Erschienen in: Multifunctional Ceramic Filter Systems for Metal Melt Filtration

Verlag: Springer International Publishing

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Abstract

Open cell ceramic foam filters are used to improve the quality of metallic cast products. They play a major role on reducing the number of inclusions within the microstructure of the cast product and restraining the liquid flow inside the mold. The newly developed carbon-bonded alumina ceramics are investigated considering the mechanical and thermal loads of the filtration process. The aim of this project is to assess the strength, the fracture mechanical behavior, and the damage properties of the filter material. Since the tiny struts of the foam have different properties than the common bulk material specimen, small specimens of millimeter size are needed. Within this work, a combination of experiments and numerical simulations are conducted for various mechanical tests at temperatures up to 1500 ${}^{\circ }\text {C}$. The first test is the small punch test (SPT), where a small disk-like specimen is supported on a circular die and loaded with a spherical tipped punch until failure of the disk occurs. The small punch fracture stress is obtained from the experiments, as well as the corresponding parameters of the Weibull distribution necessary for the evaluation of the cold modulus of rupture (CMOR). Furthermore, a modified version of the SPT, the so called ball on three ball test (B3B), is carried out. In the B3B test, miniaturized disk-shaped specimens are supported with three spherical balls and loaded with a spherical tipped punch until failure. Hereby, the fracture mechanical parameters, such as the fracture toughness, are identified with the help of numerical simulations. Moreover, the Chevron notched beam test (CNB) is used to determine the fracture toughness of the ceramics. The final test is the Brazilian disc test (BDT), where compressive loads are exerted on the specimen leading to tensile stresses along the specimen diameter, perpendicular to the applied load direction. This test is employed to determine the Young’s modulus, the yield stress and the fracture toughness. In general the simplicity of all these tests, their setup, application, and results evaluation, in addition to the ease of specimens production, make them very attractive. The size of these specimens is about one order of magnitude smaller than that of common standard tests. The accompanying numerical simulations are inevitable to extract from the measurements the wanted mechanical properties.

23.1 Introduction

Open cell ceramic foam filters are utilized during metal melt filtration processes, because they offer an active and reactive contribution to clean the metal melt. Additionally, the filters reduce melt turbulence, which also enhances the quality of the cast products due to less casting defects according to a calm melt flow. The integrity of the filters is an important requirement for industrial applications. The filters have to withstand the mechanical and thermal loads without any failure to avoid impurities of the cast product.

Within the CRC 920 new refractory materials are developed [1, 2], which have an improved thermal shock resistance. The various test methods described in this work are not necessarily restricted to certain materials. But to compare the different methods we restrict ourselves to the classes of materials developed within the CRC 920. In particular, this is carbon bonded alumina Al$_2$O$_3$-C in different chemical compositions. Here, the experimental methods are used to judge, which composition or heat treatment leads to superior properties of the material with respect to its application.

Small scale specimen testing techniques [3] are especially used if the amount of available material is limited, or if the structures made of the specific material are small. The ceramic bulk materials used for the purpose of metal melt filtration show a distinct size effect. Within large samples different properties may show up depending on the location within the samples, depending on production strategy or heat treatment. Ceramic filters used for metal melt filtration are spatial networks of struts having a diameter between approximately 100 to 2000 $\mu $m. Therefore, specimens having similar dimensions should be used to determine the properties of these filter materials.

The small punch text (SPT) is a promising testing technique for fine-grained ceramic materials [4]. In this test a small disc shaped specimen is placed on ring-like bearing and deformed by a spherical tipped punch. Depending on the material a wide range of properties can be determined [5‐7]. The SPT is applicable to determine parameters of simple failure criteria up to parameter sets for complex elastic plastic damage models [8].

The ball on three ball (B3B) test is a special variant of the SPT, where the specimen is supported by three balls instead of a ring-like support. It is especially useful for testing brittle materials or specimens having not perfect plane surfaces [9‐11].

The chevron-notched beam (CNB) test is a standardized method to evaluate the fracture toughness of ceramic materials [12‐14]. The advantage of this test is that no sharp pre-crack has to be prepared because the crack is forming itself during loading at the beginning of the test. Furthermore, no crack length measurement is required and a stable crack growth can be guaranteed due to a suitable geometry of the notch [15]. The challenge here is to find suitable geometries for sub-sized CNB specimen and their application at high temperatures.

The Brazilian disc test (BDT) utilizes cylindrical specimen, which are compressed along their vertical diameter with increasing load until specimen failure [16]. This test has been developed for brittle rock like materials [17], where grains are rather large compared to the fine grained ceramics within the scope of the CRC 920. The advantage of this test method is that no specimen deformation measurement is necessary. To obtain the material strength only the compression force at failure has to be known. The challenge is to develop a small scale version of this test and its application at high temperatures.

The finite element method is an universal tool to analyze stress and strain fields in structures under arbitrary thermo-mechanical loads. In the context of small scale specimen testing techniques it is especially useful to analyze non-standard test geometries with respect to their suitability to determine strength and fracture toughness of brittle materials under high temperature conditions. From the resulting stress fields simple failure criteria can be easily evaluated. For the analysis of fracture toughness or damage processes special tools have been developed, as the J-integral method [18‐20] or cohesive zone approaches [21, 22].

The aim of this contribution is to give an overview about the different experimental and numerical methods to determine the thermo-mechanical properties of refractory ceramics, especially those used for ceramic filters. The results obtained using the different testing techniques will be compared to each other with respect to accuracy and complexity of the test setups.

23.2 Theoretical Foundation

Within this section theoretical approaches are discussed that allow the evaluation of the strength of ceramic filter materials under thermo-mechanical loading. This includes simple classical failure criteria as well as more advanced approaches as fracture and damage mechanics including cohesive zone models.

23.2.1 Failure Criteria

Failure criteria are used to distinguish whether a material can still withstand a certain load or not [23]. The maximum stress criterion assumes that a material fails when the maximum principal stress $\sigma _1$ in a material element exceeds the uniaxial tensile strength $\sigma _{\text {t}}$ or if the absolute value of the minimum principal stress $\sigma _3$ is less than the uniaxial compression strength $\sigma _{\text {p}}$ of the material

$$\begin{aligned} -\sigma _{\text {p}} < \sigma _3 < \sigma _1 < \sigma _{\text {t}}\,. \end{aligned}$$

(23.1)

The Mohr-Coulomb failure criterion describes the critical state of a material with respect to shear as well as normal stress.

$$\begin{aligned} \frac{M+1}{2} \max \left[ \right. \left| \sigma _1 - \sigma _2 \right| + K (\sigma _1 + \sigma _2),\, \left| \sigma _1 - \sigma _3 \right| + K (\sigma _1 + \sigma _3),\quad &\nonumber \\ \left| \sigma _2 - \sigma _3 \right| + K (\sigma _2 + \sigma _3) \left. \right] & < \sigma _{\text {p}}\,, \end{aligned}$$

(23.2)

with

$$\begin{aligned} M=\frac{\sigma _{\text {p}}}{\sigma _{\text {t}}} \quad \text {and} \quad K=\frac{M-1}{M+1}\,. \end{aligned}$$

(23.3)

Especially brittle materials show a scatter of failure stresses. Therefore a statistical description of the failure processes is helpful. The Weibull theory [24] assumes that brittle materials contain randomly dispersed defects. Due to the weakest-link effect, these defects act as crack initiation points, which can cause total failure of the specimen. This theory includes a size effect, because as larger a sample is, as more likely it is to contain a critical defect. The failure probability of the Weibull theory is given as

$$\begin{aligned} P_\textrm{f}(\sigma _\mathrm {}) = 1-\exp {\left[ -\frac{V_\textrm{eff}}{V_0}\left( \frac{\sigma }{\sigma _0}\right) ^m \right] }\,, \end{aligned}$$

(23.4)

where m and $\sigma _0$ represent the Weibull modulus and the Weibull reference stress, respectively. The Weibull modulus is a measure to express the amount of scatter, whereas $\sigma _0$ corresponds to the strength of the material assuming that the probability of failure is 63.2% considering its volume.

Moreover the size effect is accounted for by the ratio of an effectively loaded volume $V_\textrm{eff}$ and a reference volume $V_0$. The effectively loaded volume is obtained by

$$\begin{aligned} V_\textrm{eff}=\int \limits _{V}\left( \frac{\sigma _\textrm{eq}}{\sigma _0}\right) ^m \textrm{d}V\,, \end{aligned}$$

(23.5)

where $\sigma _{\text {eq}}$ is an appropriate equivalent stress measure. The particular criteria used is based on the principle of independent action (PIA), which assumes that the principal stresses act independently of each other, whereby only positive values are taken into account

$$\begin{aligned} \sigma _\textrm{eq}=\root \textrm{m} \of {\langle \sigma _{I}\rangle ^m + \langle \sigma _{II}\rangle ^m + \langle \sigma _{III}\rangle ^m} \quad \text {with} \quad \langle \bullet \rangle = \frac{ \left| \bullet \right| + \bullet }{2}\,. \end{aligned}$$

(23.6)

To estimate the Weibull parameter the maximum likelihood (ML) procedure is the preferred method. Details about the ML method are given by Soltysiak and Zielke et al. [25].

A failure criterion for crack growth is defined according to the theory of linear elastic fracture mechanics. Griffith’s theory defines that a critical stress $\sigma _{\text {c}}$ is needed to propagate a crack with length a

$$\begin{aligned} \sigma _{\text {c}}=\sqrt{\frac{E G}{\pi a}}\,, \end{aligned}$$

(23.7)

where E denotes the Young’s modulus of the material and $G=2\gamma $ an energy which for brittle materials is equal to the specific surface energy $\gamma $ of the two crack faces. Another suitable criterion is the stress intensity factor (SIF) (for mode I)

$$\begin{aligned} K_I = \sigma \sqrt{\pi a}\, \end{aligned}$$

(23.8)

which can be compared to the fracture toughness value $K_{I\text {c}}$ of the material. Fracture occurs if the stress intensity factor reaches the critical fracture toughness value $K_I = K_{I\text {c}}$. Using finite element simulations SIFs can be determined via the J-integral.

$$\begin{aligned} K_I=\sqrt{J_I E'} \quad \text {with} \quad E'=\frac{E}{1-\nu ^2} \end{aligned}$$

(23.9)

for plane strain or $E'=E$ for plane stress conditions, respectively. For the evaluation of the J-integral or stress intensity factors several computational methods are available [19]. The critical mode I fracture toughness is defined as

$$\begin{aligned} K_{I\text {c}}=Y \sigma _{\text {c}} \sqrt{\pi a}\,, \end{aligned}$$

(23.10)

where Y is a dimensionless factor depending on the specimen geometry and loading conditions.

23.2.2 Cohesive Zone Model

The cohesive zone model (CZM) approach is a phenomenological description of damage and its evolution in structural components [22, 26]. CZMs can capture the failure of brittle, ductile or viscoplastic materials. The CZM is a damage model which accounts only for failure by the separation of the material along a given internal surface. CZMs can be combined with arbitrary material models for the adjacent bulk regions. In contrast to continuum damage models the potential crack path must be known from experimental investigations or other numerical analyses. The mechanical behavior of a cohesive zone is described by a traction-separation relation, also called cohesive law.

In general, the cohesive traction separation laws can be described with the following independent model parameters: the cohesive strength $t_0$, which corresponds to the maximum of the cohesive traction-separation curve, the separation work $\Gamma _0$ (i.e. fracture energy), and an internal length $s_0$, which defines the separation at the maximum traction. For a bi-linear traction separation law as shown in Fig. 23.1 alternatively a separation at total failure $s_\text {f}$ could be defined. The specific separation work $\Gamma _0$ is denoted as the area under the cohesive traction-separation curve, i.e.,

$$\begin{aligned} \Gamma _0 = \int _{0}^\infty t_n \text {d} s_n\,. \end{aligned}$$

(23.11)

Furthermore, the gradient $E_{\text {cz}}=t_0/s_0$ defines the stiffness of the cohesive zone, which describes the initial slope of the curve in the case of a bi-linear law. In the bi-linear cohesive stress-separation curve, damage of the cohesive zone $D_{\text {cz}}$ initiates upon reaching the maximum separation stress $t_0$ with the corresponding separation length $s_0$. Once the separation length $s_\text {f}$ is reached, the damage $D_{\text {cz}} = 1$ and the cohesive element is supposed to hold no more traction. It should be mentioned that many different formulations of cohesive laws are available [26], with a large variety of formulations for damage evolution. In the current work the simple bi-linear approach is used. The fracture toughness and the separation work are related via Equation (23.9), with the J-Integral replaced by $\Gamma _0$.

$$\begin{aligned} K_{I\text {c}}=\sqrt{\Gamma _0 E'} \end{aligned}$$

(23.12)

23.3 Experimental and Simulation Methods

In this section the experimental and simulation methods are described, which are used to determine the thermo-mechanical properties of the materials under consideration.

23.3.1 Material and Specimen Preparation

Carbon-bonded alumina filters, are used for molten steel filtration due to their high thermal shock resistance. As the main component of the Al$_2$O$_3$-C, 99.8$\%$-pure alumina (Martoxid MR 70, Martinswerk, Germany, $d_{90} \le {3.0\,\mathrm{\upmu \text {m}}}$) is used. In addition, the carbon content for the slurry is provided from three different sources; modified coal tar pitch powder (Carbores^® P, Rütgers, Germany, $d_{90} \le {0.2\,\mathrm{\text {m}\text {m}}}$), fine natural graphite (AF 96/97, Graphit Kopfmühl, Germany, 96.7 wt$\%$ carbon, 99.8 wt$\% \le {40\,\mathrm{\upmu \text {m}}}$), and carbon black powder (Luvomaxx N-911, Lehmann & Voss & Co., Germany, $\ge $ 99.0 wt$\%$ carbon, > 0.01 wt$\%$ ash content, primary particle size of $200 - {500\,\mathrm{\text {nm}}} $). The combination of these sources leads to a final product with 30% carbon content in its final composition. Three different material compositions with a varying amount of Carbores^® P were produced. Table 23.1 shows the main chemical composition. Also different specimen manufacturing routes are investigated, where the material block is either pressed or cast into a mold and dried. For the special material variant AC20 the detailed chemical composition is given in Table 23.2.

Table 23.1

Chemical composition of the investigated Al$_2$O$_3$-C materials. The amount of additives is related to the solid content of the material

	Pressed			Casted
wt%	AC10P	AC15P	AC20P	AC10C	AC15C	AC20C
Al$_2$O$_3$	66	66	66	68	67	66
Carbores^® P	10	15	20	10	15	20
Graphite	13	10	8	12	10	8
Carbon black	11	9	6	10	8	6
Additives	7.3	7.3	7.3	1.9	1.9	1.9

Table 23.2

Detailed chemical composition of the Al$_2$O$_3$-C slurry for an AC20 material

Raw material	% Mass fraction	Additives*	% Mass fraction
Martoxid MR 70	66.0	Castament VP 95 L	0.3
Cabores^® P	20.0	Contraspum K 1012	0.1
AF 96/97	7.7	C12C	1.5
Luvomaxx N-991	6.3	*Related to total solid content

To prepare the specimens for the applied tests, first, the slurry is poured or pressed into cylindrical or cuboid blocks ($25 \times 25 \times {150}\,{\textrm{mm}^3}$). The green bodies are then coked under reducing conditions at temperatures up to 800 or ${1400\,\mathrm{{}^{\circ }\text {C}}}$. The corresponding coking regimes are shown in Fig. 23.2. For the first variant, the heating rate is ${1\,\mathrm{\text {K}\text {min}^{-1}}}$. Starting from ${100\,\mathrm{{}^{\circ }\text {C}}}$ the temperature is increased, after an increase of ${100\,\mathrm{{}^{\circ }\text {C}}}$ the temperature is kept constant for 30 min. When the final coking temperature of ${800\,\mathrm{{}^{\circ }\text {C}}}$ is reached, a holding time of 180 min is specified. The second variant is characterized by a heating rate of ${1\,\mathrm{\text {K}\text {min}^{-1}}}$ to ${300\,\mathrm{{}^{\circ }\text {C}}}$. After a holding time of 60 min, the heating rate is increased to ${3\,\mathrm{\text {K}\text {min}^{-1}}}$ until the final temperature of ${1400\,\mathrm{{}^{\circ }\text {C}}}$ is reached. Subsequently, a holding period of 300 min finalizes the heat treatment. At a temperature of about ${235\,\mathrm{{}^{\circ }\text {C}}}$, Cabores^® P begins to melt and the non-aromatic hydrocarbons are broken up. With increasing temperature the formation of a liquid-solid mesophase begins, which solidifies at about 500 to ${550\,\mathrm{{}^{\circ }\text {C}}}$ and determines the structure of the later residual coal shock. The modified coal tar pitch therefore acts not only as a carbon source, but also as a binder. The ceramic bonding is thus ensured via carbon content. This is possible because pitches have a good bonding to both oxides and burrs. A shrinkage allowance is necessary because a volume shrinkage of about 6$\%$ occurs during the heat treatment. For the SPT and B3B samples, the cylindrical castings are turned to rods with a final diameter of $d = {8}\,{\textrm{mm}}$. From the rods discs are cut to the nominal thickness of $t = {0.5}\,{\textrm{mm}}$ using a cutting device. For the CNB samples, the cuboid bodies are separated and also brought to the final dimensions using an abrasive belt (P100 grit, 162 $\upmu $m). To cut the notch for the CNB specimens a precision diamond wire saw (Well, Mannheim, Germany) using a wire with diameter $d={0.3}\,{\textrm{mm}}$ (Well, Type A3-3) is utilized. The BDT samples are drilled with hollow diamond drilling bits from the cuboid blocks. Various sample diameters from 9.8 mm up to 15.8 mm were obtained using different drilling bits. To obtain the final disc thickness, the cylinders are then cut using a cutting device.

23.3.2 Small Punch Test

The testing device used for the small punch test and its components are integrated into a universal testing machine Inspekt Table 10 kN from Hegewald & Peschke, Germany. It is equipped with a furnace, which allows material testing up to ${1200\,\mathrm{{}^{\circ }\text {C}}}$. The furnace has three zones, which are controlled separately. To monitor the specimen temperature an additional thermocouple can be attached through a drilled hole in the apparatus. The loading force is applied via the punch inside the SPT apparatus through an upper linkage, which is directly connected to the load cell of the testing machine. Different load cells can be used depending on the expected measuring range. The measuring ranges of the load cells are ${100}\,\textrm{N}$, ${500}\,\textrm{N}$, ${1}\,\textrm{kN}$ and ${10}\,\textrm{kN}$. The load cell is located outside the furnace and is actively cooled by a fan to maintain the allowable temperature operating range. Argon as inert gas can be fed into the specimen chamber through the lower pressure linkage to prevent oxidation of the sensitive materials. The components of the SPT apparatus are shown in Fig. 23.3 and are made either of high-temperature steel (1.4841), or of 99,7% pure alumina. The lower housing (c) is attached to the compression linkage and thus forms the connection to the testing machine. The upper housing (b) serves on the one hand to guide the punch (a) and on the other hand to prevent the inert gas from escaping. In this design, the force is applied by a punch and a ceramic loading ball (e) with a diameter $d={2.5}\,{\textrm{mm}}$. The specimen (i) with a diameter $D={8}\,{\textrm{mm}}$ and thickness $t={0.5}\,{\textrm{mm}}$ is supported by the lower die (h) and centered by a ring (f). Another ring (d) is used to guide the loading ball (f) and the punch. The lower die has an inner hole with a diameter $d={4}\,{\textrm{mm}}$ and a chamfer edge with $r={0.5\,\textrm{mm}}\times {45\,\mathrm{{}^{\circ }}}$, which results in a support radius $R_\text {a}={2.5}\,{\textrm{mm}}$. This multi-part design has the advantage that parts can be changed individually if they are worn out or somehow damaged.

For small deflections of the specimen as it is expected for brittle ceramic materials Börger et al. [9] carried out elastic stress analysis and defined an expression for the failure stress depending on the maximum load $F_{\max }$ and geometric measures of the SPT setup.

$$\begin{aligned} \sigma _{\text {f}}=k\frac{F_{\max }}{t^2}=k \sigma _{\text {SP}}\,, \end{aligned}$$

(23.13)

with

$$\begin{aligned} k=\frac{3(1+\nu )}{4 \pi } \left[ 1+2 \ln \frac{R_{\text {a}}}{b} + \frac{1-\nu }{1+\nu } \left( 1 - \frac{b^2}{2 R_{\text {a}}^2} \right) \frac{R_{\text {a}}^2}{R^2} \right] \,, \end{aligned}$$

(23.14)

where R denotes the specimen radius, $R_{\text {a}}$ the support radius, t the specimen thickness, $\nu $ the Poisson’s ratio of the specimen material and, b the radius of the contact region between specimen and punch. For the contact radius b in Equation (23.14) different approximations can be found in literature [25]. The dimensionless parameter k can be seen as an empirical function [9], which can be determined by finite element simulations depending on the loading force F and Young’s modulus E of the specimen.

23.3.3 Ball on Three Balls Test

The ball on three balls (B3B) test is used to determine the biaxial flexural strength and the material parameters of the cohesive model. In the developed test apparatus, as shown in Fig. 23.4, a disc-shaped specimen with a diameter of $d={8}\,{\textrm{mm}}$ and a thickness of $t={0.5}\,{\textrm{mm}}$ is supported by three balls and loaded centrically by one ball. The apparatus can be implemented in the same test rig used for SPT for testing temperatures up to $T={1000\,\mathrm{{}^{\circ }\text {C}}}$. A second test rig using a different furnace is used for higher temperatures up to $T={1400\,\mathrm{{}^{\circ }\text {C}}}$. The test apparatus is mounted on the lower linkage of the test rig. The three ceramic support balls (h) are axially aligned by a cage (g). The specimen centering ring (f) ensures that the specimen (i) is correctly positioned. A second centering ring (d) guides the loading ball (e) with a diameter of $d={2.5}\,{\textrm{mm}}$. The punch (a) guided by the upper housing (b) is transmitting the force from the upper pressure plate of the testing machine to the loading ball. Argon as inert gas is introduced through a hole in the lower linkage of the testing machine to prevent decarburization of the specimen during testing at high temperatures. All of the mentioned parts are made of 99.7$\%$ pure aluminum oxide. The punch loads the specimen displacement-controlled at a constant displacement rate of $\dot{u}={0.05}\,{\textrm{mm}\,\textrm{min}^{-1}}$ until specimen failure occurs. The applied force is recorded by means of a load cell. As for the SPT the Equations (23.13) and (23.14) can be used to determine the stress at failure [9].

23.3.4 Chevron Notched Beam Test

To determine the fracture toughness $K_{\text {Ic}}$ of a brittle material, a four-point bending test with chevron-notched specimens is used. Different configurations of chevron-notched specimens are available in the literature [13, 15, 29]. The triangular-shaped chevron notch is cut into the specimen using a wire saw. It offers the following advantages: First, no initial pre-crack is necessary because a sharp crack develops during loading starting from the tip of the notch. Furthermore, no crack length measurement is required because the maximum force is sufficient to calculate the fracture toughness after failure of the specimen. For the subsequent investigations, the crack length a and the chevron notch parameters $a_0$ and $a_1$ are normalized by the specimen height W, resulting accordingly in $\alpha =a/W$ and $\alpha _0 = a_0/W$ and $\alpha _1=a_1/W$, respectively. Figure 23.5a shows the front view of a specimen with chevron notch as well as the geometrical quantities from the resulting sectional view, where the section plane corresponds to the notch plane. The outer specimen dimensions are $L={25}\,\textrm{mm}$, $B={5}\,\textrm{mm}$ and $W={6}\,\textrm{mm}$. Outer and inner span of the support rollers are given as $S_1=20\,\textrm{mm}$ and $S_2=10\,\textrm{mm}$, respectively. Figure 23.5b is representative of the crack length a, chevron tip dimension $a_0$, chevron dimension $a_1$ and length of crack front b. In the special design used by Zielke et al. [30, 31], one roller of loading and supporting span is replaced by a ball. Both rollers (m) and balls (k) have diameters of $d_{\text {s}}={5}\,\textrm{mm}$. A cage structure is assembled to the lower sealing of the testing machine (j). This cage structure was constructed in order to ensure the alignment of the specimen (l) as well as the loading and bearing components. The cage structure is guided by a housing (d) and consists of lower (h), middle (f), upper cage ring (c), and specimen positioning bolts (g). Upper and lower spacers and guiding rods, which are not visible in Fig. 23.6, provide the correct distance between the cage rings. The load from the upper punch (a) is transferred to the inner ball-cylinder pair via loading plate (e) and ball (b).

23.3.5 Brazilian Disc Test

The BDT is a convenient test method for brittle materials. In this test, in-plane compressive forces are applied to the disc-shaped specimens, which cause the formation of in-direct horizontal tensile forces along the vertical center-line of the specimen. The test is considered to be valid, if a central crack is formed along the vertical disc axis and split the disc into two halves. However, excessive stresses are formed along the contact lines as the compressive force increases. Due to the porous structure of the material, high amount of deformation is observed in the vicinity of the contact-line. Therefore, various test configurations were studied at room temperature to obtain a suitable method which yields valid test results. Another criterion regarding the test variants is the suitability for high temperature testing. The test variants are illustrated in Fig. 23.7. Additionally, to eliminate the false vertical displacement reading caused by the contact area deformation, surface displacement map of the sample was recorded throughout the test using an ARAMIS adjustable digital image correlation (DIC) system. This system consists of two symmetrically focused cameras which can be horizontally adjusted. The surface components are defined by GOM Correlate software which processes the digital images obtained from the cameras. The surface components are marked with the random speckle patterns created by sprayed paint droplet patterns.

A Shimadzu AGS-10 universal testing machine is used for the room temperature tests. The tests are conducted with a constant punch displacement rate of ${0.1}\,\textrm{mm}\,{\textrm{min}}^{-1}$. To observe the crack initiation sequence a high speed camera (MakroVis) is used. Apart from the vertical crack caused by tensile forces at the center, in all of the test variants symmetrical secondary arc-shaped cracks initiating laterally at the contact points were observed. With the help of high speed camera, failure of the disc is recorded for each of the test variants. Ultimately, to apply the BDT at high temperatures, a test configuration is established. The test setup is shown on the right side of Fig. 23.4. Based on the room temperature test configurations, variant from Fig. 23.7e is selected for high temperature testing. To conduct the high temperature testing, the specimen size is further reduced to a diameter of ${9.8}\,\textrm{mm}$ and a thickness of ${6.0}\,\textrm{mm}$ with a central hole of ${1.5}\,\textrm{mm}$ in diameter. To prevent oxidation of the carbon content at higher temperatures, argon gas flow is provided into the test chamber. Applied force vs. punch displacement for each of the specimens is obtained at 1200 ${}^{\circ }\text {C}$. Due to the contact area damage and displacement of the test setup, displacement data for high temperature testing is not considered to be accurate. Therefore, to obtain the material parameters, cohesive zone modeling was implemented (Fig. 23.8).

23.3.6 Finite Element Analysis

The finite element method (FEM) is used to analyze all test setups numerically. For the SPT an axisymetric model is used where the specimen is meshed with axisymetric 8-node quadratic elements. The receiving die, downholder and punch are modeled using rigid surfaces. Receiving die and downholder are fixed in space. The punch can only move in vertical direction. The contact between the rigid parts and the specimen is realized using the surface to surface interface of Abaqus. A Coulomb friction model with a friction coefficient $\mu =0.1$ completes the computational setup.

For the B3B test a 3D-model (see Fig. 23.9) is necessary, but the threefold rotational symmetry is taken into account to reduce the computational effort. The specimen is meshed with quadratic 20-node brick elements. The balls are modeled as rigid surfaces and are fixed in space, except of the loading ball, which can be moved in vertical direction. Also here, the contact between specimen and balls is realized as in the SPT model.

In order to simulate the crack growth in the chevron-notched specimens, an FE model (see Fig. 23.10) with appropriate boundary conditions was developed. 20-node hexahedral continuum elements (C3D20) with quadratic displacement functions are used, with the element edge length being 0.2 mm. An isotropic, linear-elastic material behavior is assigned, which only requires the two material parameters modulus of elasticity E and Poisson’s ratio $\nu $. Using the symmetry properties and simplifying the loading bodies, the model can be reduced to a quarter model with the corresponding symmetry boundary conditions. The support and loading rollers are modeled as rigid bodies and their contact with the sample is simplified as being frictionless. This assumption is permissible due to the rolling bearing of the loading bodies, since these minimize the frictional stresses. Furthermore, the notch radius is defined according to the diameter of the wire saw with $r_K$ = 0.15 mm, with 15 nodes along the quadrant. The contact stiffness was defined using a non-linear formulation, where the contact pressure increases exponentially as two contact surfaces approach each other. In this way, roughness and surface waviness of the contact surfaces, which arise due to production, can be taken into account and the run-in characteristics of the measured curves can thus be mapped.

The cohesive elements (COH3D8) have a rectangular base and a linear displacement approach, with element edge lengths ranging from 0.015 to 0.14 mm. Since the cohesive elements are located on a symmetry plane of the model, there are conditions for maintaining the symmetry. The studied parameters of the cohesive model are cohesive stiffness $E_{\text {cz}}$, cohesive strength $t_0$, work of separation $\Gamma _0$, and modulus of elasticity E. Due to its negligible effect the Poisson’s ratio was assumed as $\nu $ = 0.2.

For the finite element analyses, of the BDT a quarter model of the disc is sufficient. At the vertical symmetry plane cohesive elements are attached with appropriate symmetry conditions. The load is applied using a rigid plane at the top. Figure 23.11 show the models for Var. A and Var. E (cf. Fig. 23.7). The stress analysis shows that for Var. E exists a small stress concentration area at the top of the hole where the crack initiates. For Var. A a rather larger area of tensile stresses along the vertical symmetry plane is observed.

23.4 Results

23.4.1 Small Punch Test

The SPT was mainly used to obtain the mechanical strength of the different materials. Figure 23.12 shows two typical load vs. deflection curves obtained from SPTs, one measured at room and the other at high temperature (${800\,\mathrm{{}^{\circ }\text {C}}}$). At the beginning the curves show an ascending slope as it is expected for a linear elastic material. After a certain point small load drops are observed which are caused by evolving cracks at the lower side of the specimen. The point where the first significant load drop appears defines the point of initial failure and the corresponding failure stress values are obtained using the Equations (23.13) and (23.14) presented in Sect. 23.3.2. The obtained failure stresses have a significant scatter, so a Weibull analysis is performed, the results of which are shown right in Fig. 23.12. The curves represent the Weibull distributions fitted to the corresponding data. Their intersection with the horizontal line corresponding to 63.2% failure probability defines the Weibull reference stress $\sigma _0$ and the slope corresponds to the Weibull modulus m. Although the ranges of failure stresses for the different test temperatures overlap, differences for Weibull stress and modulus can be statistically detected. The Carbores^® P binder content has a significant influence on the strength of the material system. To investigate this influence, the Carbores^® P content of the carbon fraction was varied from 10 wt% to 20 wt%. In the case of the cast material the total carbon content after coking was fixed to 30 wt% for all systems. The result of this investigation is shown left in Fig. 23.13 in the form of a Weibull diagram. It can be seen that, the strength of the cast material system increases with increasing binder content. That is, a larger proportion of coal tar pitch leads to a higher strength of the overall system. However, it can be stated that the binder content has no significant influence on the Weibull modulus. This means that the probability of having a defect with failure-relevant size and location in the microstructure is not influenced by the binder content. This can be linked to the melting of the binder during fabrication.

The materials were also evaluated according to three different manufacturing processes. Prior to drying and coking, the slurries were casted (AC20C), isostatically (AC20Pi) or uniaxially (AC20Pu) pressed to $p={150\,\mathrm{\text {M}\text {Pa}}}$. It can be clearly seen on the right side of Fig. 23.13 that the strength of the cast material is lower than the strength of the pressed materials. Uniaxial pressing yields a higher strength of the material with lower scatter than isostatic pressing.

To study the effect of the heat treatment on the material strength, the casted slurries containing 20 wt% Carbores^® P were coked at temperatures of ${800\,\mathrm{{}^{\circ }\text {C}}}$ (AC20C) and ${1400\,\mathrm{{}^{\circ }\text {C}}}$ (AC20T). Figure 23.14 illustrates the difference of these two heat treatments. The rhombohedral phase of graphite is metastable above ${1300\,\mathrm{{}^{\circ }\text {C}}}$. Therefore, a transformation of this phase takes place. In addition, a longer holding time in combination with an elevated temperature leads to an increased graphitization of the coal tar pitch. The higher coking temperature does not exert any significant influence on the Weibull modulus. However, it can be seen that the Weibull stress is lower for the material coked at a higher temperature. This can be explained by the increased crystallization of the carbon. As the degree of crystallization in the carbon increases, the anisotropy of the mechanical and thermal properties also increases. This results from an increasing regularity in the arrangement of graphene planes. In addition, the removal of impurity atoms between the planes leads to the formation of additional $\pi $-bonds, which have a very low binding energy. All these defects facilitate sliding of the graphene planes parallel to the planes. Macroscopically, this can be measured as reduced strength of the material. Based on this investigation, it is clear that the carbon has a significant influence on the strength of the material system. This depends not only on the binder content, but also on the chemistry of the carbon. A lower degree of order of the graphite planes has a positive effect on the strength of the material system.

Soltysiak et al. [34] also investigated the influence of the microstructure on the fracture behavior of carbon bonded alumina. Not all specimen show significant load drops in their load-deflections curves. The failure of some specimen manifests itself just as a decrease in the slope of the load-deflections curves. It was found that the load drops are caused by the breakage of rather large Carbores^® P grains near the specimen surface. The absence of large Cabores^® P grains is leading to a more continuous fracture of the specimen.

23.4.2 Ball on Three Ball Test

Zielke et al. [35] compared results obtained from SPTs and B3B tests on a AC20C material. The B3B tests were done with different support radii $R_{\text {a}}$. The failure stresses are obtained similar as to the SPT procedure. For each variant 30 specimen were tested and a Weibull analysis performed. The results are summarized in Table 23.3 and show a good agreement between all different test setups. The support radius has no significant influence on the obtained Weibull parameters.

Table 23.3

Comparison of identified Weibull parameters using B3B tests with varied support radius and SPTs. The values in brackets correspond to 95% confidence interval [35]

	$R_{\text {a}}$ [mm]	$\sigma _{0}$ [MPa]	m
B3B	2.31	21.74 (20.01, 23.67)	4.88 (3.57, 6.56)
B3B	2.89	21.35 (19.66, 23.25)	4.88 (3.57, 6.57)
B3B	3.46	20.56 (19.02, 22.28)	5.18 (3.79, 6.97)
SPT	2.50	19.06 (17.61, 20.57)	5.29 (3.88, 7.13)

Table 23.4

Comparison of identified Weibull parameters using B3B tests for different manufacturing routes. The values in brackets correspond to 95% confidence interval [25]

Specimen	N	$\sigma _{0}$ [MPa]	m	T [ ${}^{\circ }\text {C}$]
Slip cast	30	18.04 (16.87, 19.26)	6.18 (4.53, 8.32)	20
Slip cast	30	25.37 (23.97, 26.81)	7.32 (5.36, 9.85)	800
Sprayed	30	20.50 (18.46, 22.69)	3.97 (2.91, 5.35)	20
Sprayed	30	24.61 (21.16, 28.52)	2.85 (2.06, 3.88)	800

Further investigations of Zielke et al. [25] using the B3B test were done with respect to different manufacturing routes and test temperatures. The slurry composition corresponds to that of a AC20C material. The results are presented in Fig. 23.15 and Table 23.4. Again the material has greater strength at high temperatures than at room temperature. But there is a significant difference regarding the scatter of the strength data. The Weibull exponent for sprayed specimen is much smaller than for cast specimen. This is because the slip cast specimens show a more homogeneous microstructure than the sprayed ones. Another important aspect regarding the strength of Al$_2$O$_3$-C is the coking temperature, which has been investigated by Zielke et al. [27]. In this research the testing facilities have been upgraded with a furnace, which enables test temperatures up to 1600 ${}^{\circ }\text {C}$. Figure 23.16 shows the main results. For test temperatures higher than 1200 ${}^{\circ }\text {C}$ the specimens show no longer neither elastic behavior nor brittle fracture. The deformation at failure increases with higher temperatures and includes some plastic or visco-plastic portions, which results in non-linear rising slopes of the load deflection curves. Figure 23.16b shows the Weibull plot for specimens coked at 800 ${}^{\circ }\text {C}$ and tested at room temperature. The most interesting effect is displayed in Fig. 23.16c. The material coked at 800 ${}^{\circ }\text {C}$ has its highest strength also at 800 ${}^{\circ }\text {C}$. Similarly, the material coked at 1400 ${}^{\circ }\text {C}$ has its highest strength at 1400 ${}^{\circ }\text {C}$. Therefrom it can be stated that the material strength has its highest values at the coking temperature. At the coking temperature during material production, the material is in a state without micro cracks and an ideal binding of the Al$_2$O$_3$ particles to the carbonaceous matrix is established. This state is achieved because of the long holding time. When the specimens are cooled down, micro cracks and pores develop between the two phases due to the different thermal expansion coefficients ($\alpha _{\text {C}} > \alpha _{\text {Al}_2\text {O}_3}$). These cracks can propagate or branch. This leads to a macroscopic measurable reduced strength. If the testing temperature increases starting at room temperature, the first small cracks between Al$_2$O$_3$ particles and carbon matrix are closed. With a further temperature increase up to the coking temperature, the cracks are closed completely and internal compressive stresses occur leading to a rising strength. If the testing temperature is higher than the coking temperature, the expanding Al$_2$O$_3$ particles induce thus tensile stresses in the carbonaceous matrix and new cracks are generated. However, these cracks do not propagate at the particle-matrix interface as before, but within the matrix. The result is a decrease in strength. Fracture toughness was determined by Zielke et al. [28] using simulations of the B3B test utilizing CZM. The underlying FE-model is shown in Fig. 23.9. To identify the cohesive parameters, an optimization approach is utilized. Hereby, an optimal parameter set $\boldsymbol{p}^*$ is found by minimizing the error norm (23.16) between the simulated and measured load deflection curves.

$$\begin{aligned} \boldsymbol{p}^*&= \arg \min \left( \phi (\boldsymbol{p})\right) \end{aligned}$$

(23.15)

$$\begin{aligned} \phi (\boldsymbol{p}) &= \frac{1}{N} \sum _{k=1}^N \left( \frac{F_{\text {sim}}(u_k,p_i)-F_{\text {exp}}(u_k)}{F_{\max }} \right) ^2 \end{aligned}$$

(23.16)

The parameter set contains the elastic modulus and the unknown cohesive parameters $\boldsymbol{p}=[E,\Gamma _0,t_0]$. The cohesive stiffness $E_{\text {cz}}$ has no influence on the result of the simulation, if it is above a certain limit. The displacement range used to compute the error norm goes from zero up to a value slightly larger as the displacement for the maximal force. Figure 23.17 shows the optimization scheme and the comparison between experimental and simulated force displacement curves. The simulation with the cohesive zone model is able to capture the load drop, where the primary cracks appear if the model is properly calibrated. Table 23.5 displays the identified values for three different B3B setups. The material investigated is a AC20 ceramic.

Table 23.5

Identified cohesive parameters and resulting fracture toughness obtained from three B3B tests with varying support radius $R_\text {a}$ [28]

$R_\text {a}$ [mm]	2.31	2.89	3.46
E [MPa]	5930	6022	8067
$\Gamma _0$ [N/mm]	0.043	0.037	0.025
$t_0$ [MPa]	31.87	36.25	33.88
$K_{I\text {c}}$ MPa$\sqrt{\text {m}}$	0.50	0.47	0.45

23.4.3 Chevron-Notched Beam Test

From the CNB test the stress intensity factor can be obtained if energy relations are considered [36]. The available strain energy of the deformed specimen and the necessary energy for crack growth must be equal. The strain energy is a function of applied load F, specimen and crack geometry, which is defined by the dimensionless parameters $\alpha $, $\alpha _0$ and $\alpha _1$. The expression for the stress intensity factor than reads

$$\begin{aligned} K_I=\frac{F}{B \sqrt{W}} \left[ \frac{E B}{2(1-\nu ^2)} \frac{\alpha _1-\alpha _0}{\alpha -\alpha _0} \frac{\text {d}C_{\text {CNB}}}{\text {d}\alpha } \right] ^{1/2} = \frac{F}{B \sqrt{W}} Y^*\,. \end{aligned}$$

(23.17)

Due to the notch shape the CNB specimen, the length of crack front increases with increasing crack depth $\alpha $. Regarding the geometry function $Y^*$, the derivative of the compliance $\text {d}C_{\text {CNB}}/\text {d}\alpha $ of the specimen also increases with crack depth $\alpha $. But, the notch geometry term $(\alpha _1-\alpha _0)/(\alpha -\alpha _0)$ decreases with increasing $\alpha $. These relations lead to the fact that the function $Y^*(\alpha )$ possesses a minimum $Y^*_{\min }$, as shown right in Fig. 23.18. At the crack length $\alpha =\arg \min (Y^*(\alpha ))$ the maximum load $F_{\max }$ is observed and the critical fracture toughness can be calculated using

$$\begin{aligned} K_{I\text {c}}=\frac{F_{\max }}{B \sqrt{W}} Y^*_{\min }\,. \end{aligned}$$

(23.18)

In the literature [29] empirical formulas can be found for $Y^*_{\max }$ based on compliance measurements and FEM simulations, which are applicable for certain ranges of $\alpha _0$ and $\alpha _1$. Numerical investigations using a cohesive zone model [31] have shown that for the given notch geometry suitable values for $\alpha _0$ and $\alpha _1$ can be found. A test is valid if the crack length at failure is $\alpha _0 < \alpha < \alpha _1$. The probability to get a valid test is best if $\alpha = 0.5 (\alpha _1-\alpha _0)$. Moreover, lower load magnitudes at failure has less impact on the deformation of the test setup. This allows having a testing machine with less stiffness. Therefore, the choice for $\alpha _1$ is close to 1 and following the right diagram in Fig. 23.18 optimal values for the initial crack length are $\alpha _0 = 0.2 \dots 0.4$. In the study done by Zielke et al. [31] 15 CNB specimen made of one block of slip casted Al$_2$O$_3$-C with $0.18 < \alpha _0 < 0.29$ and $0.89 < \alpha _1 < 0.97$ were tested. 14 of these specimens exhibited a valid load-displacement curve as well as a valid notch geometry and an average fracture toughness value of $K_{I\text {c}}=0.576 \pm 0.037$ MPa$\sqrt{\text {m}}$ was obtained.

23.4.4 Brazilian Disc Test

The variant of the BDT with the small hole is illustrated in Fig. 23.19. The random surface speckle pattern which is used for the DIC measurement is shown in Fig. 23.19a. In Fig. 23.19b shows the specimen after failure, where the central initial and two symmetrical secondary cracks are presented. Figure 23.19c shows snap shots from the high speed cameras and illustrates the growth of the initial crack starting from the hole of the disc.

Using the surface displacements acquired through DIC (see Fig. 23.20a), the displacement output from the simulation is fitted to the experimental values (Fig. 23.20b). For this, elastic parameters are the subject of iteration. Despite differences in sample and punch geometries for each of the variants, similar values for the elastic parameters are obtained [32]. The combined analysis DIC and FEM, provides such an accuracy in calculation of the elastic parameters. The average values for $E=15$ MPa and $\nu =0.26$ are achieved for Al$_2$O$_3$-C at room temperature (Table 23.6). Moreover to illustrate the formation of tensile stresses along the loading diameter of the specimen, the stress plot of a specimen before failure is given in Fig. 23.20c.

Table 23.6

Elastic parameters at room temperature identified by each of the test variants [32]

Setup	E [MPa]	$\nu $ [–]
Var. A	17.1	0.23
Var. B	15.2	0.28
Var. C	13.8	0.22
Var. D	14.1	0.36
Var. E	14.4	0.25
Average	14.9	0.27

Since DIC is not applicable at high temperature tests, and due to priorly mentioned contact area problems, an accurate reading of the sample deformation is not possible. Therefore, the only reliable test result is the applied force on the specimen. Since the location of the crack formation is well-predicted for specimens with holes, this sample geometry allows the application of cohesive zone modeling. To identify the material parameters at high temperatures measured force data is used in models to investigate cohesive parameters of the material. Using the optimization scheme shown in Fig. 23.17, elastic modulus and cohesive zone parameters could be obtained. Thus, determined high temperature cohesive parameters for Al$_2$O$_3$-C are given in Table 23.7. To find out the critical fracture toughness, Equation (23.12) is used. Using the parameters given in Table 23.7, simulation results are in good agreement with the experimental data as shown in Fig. 23.21. The obtained fracture toughness is in good agreement with those obtained from B3B and CNB tests.

Table 23.7

Identified cohesive parameters and resulting fracture toughness obtained from BDT test at high temperature

E [MPa]	$\Gamma _0$ [N/mm]	$t_0$ [MPa]	$K_{I\text {c}}$ [MPa$\sqrt{\text {m}}$]
7350	0.038	26.4	0.55

23.5 Conclusions

Small scale specimen techniques are suitable methods to determine the temperature-dependent fracture and damage properties of ceramic filter materials. Regarding brittle ceramics the B3B test should be the preferred method, since it provides a proper three point support of the specimen, even if their surface is not perfectly flat. The SPT has a slightly simpler setup but can be applied only if the specimen have a very precise geometry, especially flatness. For both SPT and B3B test simple equations can be used to determine material strength properties from measured critical loads at failure. Due to the large scatter of the experimental data for the brittle materials Al$_2$O$_3$ and Al$_2$O$_3$-C a larger number of specimens should be tested. The minimum number of required tests is about 25–30 to reach proper confidence intervals. Both tests have been applied successfully to find an optimal chemical material composition as well as suitable production and heat treatment procedures. The experimental costs for both tests are rather low, due to the small amount of material necessary, relatively small testing machines and small furnaces, which can be heated up and cooled quickly and requiring only a small amount of energy.

The CNB test is especially designed to obtain fracture toughness values for brittle ceramics. It yields repeatable experimental results. Although the specimen preparation requires the additional step for cutting the notch, it is not too difficult to obtain. However, the test setup is much more complex than that of SPT or B3B tests and contains multiple small-sized and fragile ceramic parts. The costs for those parts are approximately ten times higher than those of a SPT or B3B tests. The cost of conducting the experiment is comparable to SPT and B3B tests, since the same testing machines and furnaces are usable.

The BDT has been tested as an alternative method to the former ones, because of its very simple test setup. Here, difficulties remain regarding the contact zones between specimen and loading plates, where the high contact stresses can cause local damage near the specimen surface. By varying the geometry of the test it was found, that small holes within the specimen lead to more repeatable results with less scatter. The costs for specimen preparation and testing are comparable to those of SPT and B3B test.

FEM simulations with cohesive zone models enable the identification of fracture toughness values from B3B, CNB or BD tests. Therefore corresponding FE-models are developed, which contain cohesive elements at the potential crack locations. With the help of optimization schemes the cohesive zone parameters can be identified, which are directly related to the fracture toughness of the tested material.

Significant data on material behavior is obtained throughout this study. To sum up, a direct temperature dependency is observed for the strength and fracture toughness of the filter materials. Linear elastic behavior of material at room temperature transforms into viscoplasticity at temperatures above 1200 ${}^{\circ }\text {C}$. While higher coking temperatures inversely affect the material strength at room temperature, the peak strength of the filter materials are observed at the vicinity of their coking temperature. There is a direct dependency of binder content and strength of filter materials.

Acknowledgements

The authors gratefully acknowledge the work and efforts of Dr.-Ing. S. Soltysiak and Dr.-Ing. H. Zielke done in the previous funding periods of the CRC 920. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 169148856—CRC 920: Multi-Functional Filters for Metal Melt Filtration—A Contribution towards Zero Defect Materials, subproject C03. Furthermore, the authors acknowledge computing time on the compute cluster of the Faculty of Mathematics and Computer Science of Technische Universität Bergakademie Freiberg, operated by the computing center (URZ) and funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 397252409.

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Titel: Determination of the Temperature-Dependent Fracture and Damage Properties of Ceramic Filter Materials from Small Scale Specimens
verfasst von: Martin Abendroth
Shahin Takht Firouzeh
Meinhard Kuna
Bjoern Kiefer
Verlag: Springer International Publishing
Buch: Multifunctional Ceramic Filter Systems for Metal Melt Filtration
Print ISBN: 978-3-031-40929-5

Electronic ISBN: 978-3-031-40930-1

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-40930-1_23

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