SPECIAL ISSUE
Microcracking in calcite and dolomite marble: microstructural
influences and effects on properties
Victoria Shushakova • Edwin R. Fuller Jr. •
Siegfried Siegesmund
Received: 10 July 2012 / Accepted: 12 September 2012 / Published online: 25 September 2012
 The Author(s) 2012. This article is published with open access at Springerlink.com
Abstract Microstructure-based finite-element analysis
with a microcracking algorithm was used to simulate an
actual degradation phenomenon of marble structures, i.e.,
microcracking. Both microcrack initiation and crack prop-
agation were characterized, as were their dependence on
lattice preferred orientation (LPO), grain shape preferred
orientation (SPO), grain size, marble composition (calcite
and dolomite) and grain-boundary fracture toughness. Two
LPOs were analyzed: a random orientation distribution
function and an orientation distribution function with strong
directional crystalline texture generated from a March–
Dollase distribution. Three SPOs were considered: equiaxed
grains; elongated grains and a mixture of equiaxed and
elongated grains. Three different grain sizes were consid-
ered: fine grains of order 200 lm (only calcitic marble);
medium size grains of order 1 mm (calcitic and dolomitic
marbles); and large grains of order 2 mm (only dolomitic
marble). The fracture surface energy for the grain bound-
aries, cig, was chosen to be 20 and 40 % of the fracture
surface energy of a grain, cxtal, so that both intergranular and
transgranular fracture were possible. Studies were performed
on these idealized marble microstructures to elucidate the
range of microcracking responses. Simulations were per-
formed for both heating and cooling by 50 C in steps of
1 C. Microcracking results were correlated with the ther-
moelastic responses, which are indicators related to degra-
dation. The results indicate that certain combinations of
LPO, SPO, grain size, grain-boundary fracture toughness
and marble composition have a significant influence on the
thermal-elastic response of marble. Microstructure with the
smallest grain size and the highest degree of SPO and LPO
had less of a tendency to microcrack. Additionally, with
increasing SPO and LPO microcracking becomes more
spatially anisotropic. A significant observation for all
microstructures was an asymmetry in microcracking upon
heating and cooling: more microcracking was observed
upon cooling than upon heating. Given an identical micro-
structure and crystallographic texture, calcite showed larger
thermal stresses than dolomite, had an earlier onset of mi-
crocracking upon heating and cooling, and a greater mi-
crocracked area at a given temperature differential. Thermal
expansion coefficients with and without microcracking were
also determined.
Keywords Marble  Microcracking  Finite-element
modeling  Lattice preferred orientation  Shape preferred
orientation  Strain energy density  Maximum principal
stress  Thermal expansion coefficient  Thermal expansion
anisotropy
Introduction
Marble has been used as decorative and constructive
material since ancient times. However, marbles can be
extremely sensitive to weathering and to degradation, and
hence can have limited durability. The Taj Mahal, the
famous masterpiece of architectural art (see Fig. 1), is a
white marble mausoleum situated in Agra, India. Due to
acid rain generated from an oil refinery and foundries, the
marble fac¸ades on the Taj Mahal have been losing their
luster, brightness, and white color into a sickly shade.
Siegesmund et al. (2000, 2007) pointed out that physical
weathering is thought to be the initial stage of marble
degradation.
V. Shushakova (&)  E. R. Fuller Jr.  S. Siegesmund
Geowissenschaftliches Zentrum der Universita¨t Go¨ttingen,
Goldschmidtstrasse 3, 37077 Go¨ttingen, Germany
e-mail: victoria.shushakova@gmail.com
123
Environ Earth Sci (2013) 69:1263–1279
DOI 10.1007/s12665-012-1995-2
Many historical marble sculptures and monuments show
clear evidence of deterioration, which is visible as a loss of
relief structure (Fig. 2a) or fac¸ade bowing (Fig. 2b). The
reason for the degradation is mainly due to climate. Many
studies have shown that increases and decreases of tem-
perature can induce these degradation phenomena in mar-
bles, e.g., Kessler (1919), Battaglia et al. (1993), Winkler
(1994), Siegesmund et al. (2000), Zeisig et al. (2002).
Calcite and dolomite, the rock-forming minerals in
marble, have a large anisotropy in their coefficients of
thermal expansion along the different crystallographic
directions. Consequently, in a polycrystalline structure the
stochastic misfit strains produced by this thermal expansion
anisotropy can produce large internal stresses, which occur
mainly at triple junctions and along the grain boundaries
(Siegesmund et al. 2000). Independently, or in combination
with environment (e.g., moisture-assisted fracture), these
stresses can result in degradation phenomena, such as
microcracking and granular disintegration (Koch and
Siegesmund 2004; Siegesmund et al. 2008). Hot-stage
Fig. 1 Representative view of the fac¸ade of the Taj Mahal (Agra, India)
Fig. 2 Two examples of marble deterioration: a the weathered column of Alhambra Granada, Spain; and b bowing of marble fac¸ade panels in
Copenhagen
1264 Environ Earth Sci (2013) 69:1263–1279
123
environmental scanning electron microscopy (ESEM) can
be used to provide direct evidence of such degradation
phenomena during thermal cycling. In Fig. 3 one can
directly observe the formation and propagation of micro-
cracks near a triple junction in marble during the first
heating and then cooling cycle. Such in situ experiments
provide great in-depth knowledge of degradation pro-
cesses, e.g., the loss of cohesion and concomitant micro-
crack formation along grain boundaries, and confirms that
the initial decay process in marbles results from tempera-
ture changes and the misfit strains that result from the
anisotropy in thermal expansion of the marble crystals.
Such thermal expansion anisotropy degrades calcitic mar-
bles more than dolomitic marbles, and the degradation is
more severe if the marbles have straight grain boundaries,
e.g., Weiss et al. (2002), Siegesmund et al. (2000), Luque
et al. (2011).
Many fabric parameters influence this thermal degrada-
tion, notably, lattice preferred orientation (LPO), shape pre-
ferred orientation (SPO), grain size, grain-boundary
toughness, grain-to-grain misorientation (Siegesmund et al.
2000; Weiss et al. 2002, 2003, 2004; Saylor et al. 2007;
Siegesmund et al. 2008; Zeisig et al. 2002; Shushakova et al.
2011) and marble composition (Royer-Carfagni 1999; Sieg-
esmund et al. 2000), as well as the single-crystal properties of
the rock-forming minerals. Additionally, as has already been
mentioned, the type of grain boundaries is an important
factor that determines the behavior of marbles during thermal
tests. Simple grain boundaries indicate a lower binding
energy between them and interlobate or seriate grain
boundaries reflect higher binding energy between grains (see
Siegesmund et al. 2000; Grelk et al. 2004).
The goal of the present study is to establish the influence
of fabric parameters on the propensity and degree of
microcracking in marble structures, and to relate the deg-
radation to microstructural indicators, such as stored
energy strain density and maximum principal stress. Two-
dimensional finite-element simulations are used to eluci-
date microcracking initiation and propagation in calcitic
and dolomitic marbles. The influence of fabric parameters
include: extremes of crystallographic texture, a random
orientation distribution function (ODF) and crystallo-
graphic texture 20 times a random ODF; three morpho-
logical types of microstructures; different grain sizes;
different grain-boundary toughness; and different marble
composition.
Fig. 3 Thermally induced
microcrack formation observed
in a hot-stage environmental
scanning electron microscopy
(ESEM) on a marble sample
upon heating and then cooling.
Sketches in the right column
illustrate schematically: a the
initial rock fabric; b the red
arrows shows the crack opening
onset during heating due to
dilatation; and c the green
arrows illustrate the crack
closure on cooling and the
resulting formation of new
cracks, when closure is hindered
Environ Earth Sci (2013) 69:1263–1279 1265
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Methods
Microstructure
Two-dimensional hypothetical rock microstructures were
used as input micrographs for the finite-element simula-
tions. They were generated by using a nucleation and
growth algorithm, which is described elsewhere (Ito and
Fuller 1993; Miodownik et al. 1999; Saylor et al. 2004,
2007; Shushakova et al. 2011). Three types of SPO were
examined: equiaxed grains; elongated grains; and a mixture
of equiaxed and elongated grains (see Fig. 4). The micro-
structural images used for this study have a resolution of
1,000 9 1,000 pixels. The number of grains in the equi-
axed, mixed, and elongated microstructures are 382 grains,
347 grains, and 312, respectively.
When fracture is included as a phenomenon in finite-
element simulations, the material now has a length scale
(namely, the fracture surface energy divided by an elastic
modulus), by which to measure grain size. Heretofore (e.g.,
Shushakova et al. 2011), outputs from microstructure-
based simulations (e.g., elastic strain energy density and
maximum principal stress) were independent of grain size;
the only units being those of strain energy density
(kJ m-3 = 10-6 GPa) or stress (GPa). Now, grain size, or
equivalently, the size of the microstructural image, is
important as it influences the parameter that is input into
the simulation for the fracture surface energy. Accordingly,
we introduce grain size as an important fabric, or micro-
structural parameter. We consider here three grain sizes:
200 lm; 1 mm; and 2 mm. However, due to the com-
plexity of three SPOs in the present simulations (e.g., two
different grain distributions in the mixed microstructure
and non-equiaxed grains in the elongated microstructure), a
better dimensional metric is the size of the microstructural
image. Thus, the grain diameters for equiaxed grains and
the grain widths for elongated grains were measured for a
representative number of grains in each SPO microstruc-
ture. Using an average of these values to compute a
nominal average grain size, the size of the microstructure
was adjusted to give the three nominal average grain sizes.
For a nominal average grain size of 1 mm, the size of the
micrograph for the equiaxed and the mixed microstructure
is 16.13 mm 9 16.13 mm, and the size for the elongated
microstructure is 13.89 mm 9 13.89 mm. Similarly, when
the nominal average grain size is 200 lm and 2 mm, the
sizes of the micrographs (edge lengths), respectively, are
3.125 and 31.25 mm for the equiaxed and mixed micro-
structure, and 2.778 and 27.78 mm for the elongated
microstructure.
The laboratory reference frame is specified in Fig. 4 by
the Cartesian vectors ei, i = 1, 2, 3. The e1- and e2-direc-
tions are in the plane of the microstructure, and generally
point perpendicular and parallel to the SPO, respectively, if
shape orientation is present. The e3-direction is out of the
plane of the microstructure and points perpendicular to the
SPO.
Crystalline texture
To assign orientations to each grain, orientation distributions
were generated via the March–Dollase distribution (Dollase
1986; Blendell et al. 2004). The applied algorithm here is
described in detail elsewhere (Saylor et al. 2007; Shushak-
ova et al. 2011). Given a texture direction, the crystallo-
graphic c-axes are distributed within a cone of half angle h
about the texture direction. The polar angle of the orienta-
tion, h, with respect to the texture direction is given by
h ¼ arccos
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Mð1  PÞ2
1 þ ð1  PÞ2ðM  1Þ
s
where M is a ‘‘texture’’ parameter that characterizes the
degree of texture and gives the value of the maximum in
Fig. 4 Microstructures used for the finite-element modeling: a an equiaxed microstructure of 382 grains; b a mixed microstructure of 347 grains;
and c an elongated microstructure of 312 grains. The coordinate system, ei, is indicated
1266 Environ Earth Sci (2013) 69:1263–1279
123
the distribution in units of multiples of random distribution
(MRD). P is a random number selected between 0 and 1.
After determining the polar angle, the two azimuthal
angles of the orientation, u and x, are selected randomly in
the range of 0–2p. Two different types of LPO were gen-
erated using M values of 1 and 20. With increasing value of
M, the orientations become more textured, i.e., if M = 1
the distribution is random, when M = 20 we have a strong
crystallographic texture.
Typical LPOs for marbles, as described in Leiss and
Ullemeyer (1999), are textured with the c-axes perpen-
dicular to the SPO. Accordingly, while there are many
ways to align LPO with respect to SPO (e.g., three mutu-
ally orthogonal idealized configurations were considered
by Shushakova et al. 2011), in the current simulations we
have aligned the c-axes perpendicular to the shape fabric
and out of plane, i.e., parallel to the e3 direction. A rep-
resentative pole figure for such texture is shown in Fig. 5
along with a pole figure for a random ODF. The pole fig-
ures show the individual poles of the crystallographic
c-axis for each grain in the microstructure. The contour
lines are selected MRD contours from the March–Dollase
distribution, from which the orientations were chosen.
Finite-element simulations
For the finite-element simulations the microstructure-based
object-oriented finite-element program (OOF), developed
at the National Institute of Standards and Technology
(Langer et al. 2001), was used. The OOF software is in the
public domain. Executables, source code, and manuals are
available at: http://www.nist.gov/mml/ctcms/oof/. The
OOF1 software was used here.
An adaptive meshing algorithm was used to create a
finite-element mesh with a grain-boundary phase that has
the same thermoelastic properties and orientations as the
host grains, but with a different fracture toughness, i.e., the
grain-boundary toughness, cig. The meshing procedure is
described elsewhere (Langer et al. 2001; Weiss et al. 2002;
Chawla et al. 2003; Saylor et al. 2007; Wanner et al. 2010).
The final mesh consisted of 140,358 triangular elements for
the equiaxed microstructure; 207,208 triangular elements
for the mixed microstructure; and 163,668 elements for the
elongated microstructure.
After creating the mesh, thermoelastic properties were
assigned to the grains and to the grain boundaries accord-
ing to which marble composition (calcite or dolomite) was
being considered. The single-crystal elastic constants for
both minerals, Cij, (Bass 1995) and the crystalline coeffi-
cients of thermal expansion for calcite (Kleber 1959) and
dolomite (Reeder and Markgraf 1986) were used (see
Table 1).
Thirty unique configurations specified by SPO, LPO,
grain size, grain-boundary toughness and marble compo-
sition were used in the simulations. Three different sets of
orientations were generated for the calcite microstructures
with equiaxed, mixed and elongated grains, random and 20
times random texture, a nominal grain size of 1 mm, and a
grain-boundary toughness of 40 % of that for the grains.
For other microstructural configurations only one replica-
tion of the ODF was used in the simulations. Thus, with
replications, 42 configurations were used as the basis for
the finite-element simulations.
Strain energy density and maximum principal stress
were calculated for all microstructure configurations on
heating and cooling by 50 C. Simulations were done with
and without allowing microcracking to occur.
The fracture surface energy of single crystals is typically
of order 0.3–1.1 Jm-2 (Becher and Freiman 1978; White
et al. 1988). Accordingly, in the present simulations the
crystalline or grain fracture toughness, cxtal, for both calcite
and dolomite was assumed to be 0.5 Jm-2, and isotropic.
Little is known about experimental values for the grain-
boundary toughness. Accordingly, to examine the influence
of the intergranular fracture resistance, the grain-boundary
toughness was assumed to be 40 and 20 % of cxtal, thus the
Fig. 5 Representative pole
figures showing individual poles
of the crystallographic c-axis for
each grain in the microstructure:
a orientations chosen from a
random ODF; b orientations
chosen from a March–Dollase
ODF with fiber texture that is 20
times random along the North–
South pole. The contour lines
correspond to MRDs from the
March–Dollase distribution
function of 16 (blue), 4 (green),
1 (orange) and  (red). c-axis
align parallel to the e3 direction
Environ Earth Sci (2013) 69:1263–1279 1267
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fracture resistance energy of the grain boundaries was
cig = 0.2 Jm
-2 and cig = 0.1 Jm
-2, respectively. For
simulations of microcrack initiation and crack propagation
a Griffith-type fracture criterion was used. This criterion is
based on the energy balance between mechanical and sur-
face energy (Griffith 1921). How it is implemented in OOF
is described in detail in Zimmermann et al. (2001) and Galal
Yousef et al. (2005). Since the elastic constants input into
OOF have units of GPa, the fracture surface energy has units
of GPa times the size of the microstructure. Thus, by varying
the input value of the fracture surface energy, one can vary
the size of the microstructure, and hence, the nominal
average grain size, as discussed above.
Fracturing in OOF is accomplished by reducing the
directional elastic moduli (both the tensile and shear
moduli) across the fracture plane by a multiplicative factors
(Carter et al. 1998; Zimmermann et al. 2001; Galal Yousef
et al. 2005). Multiplicative factors were chosen to be
1 9 10-6 and 2 9 10-6 in this study.
The finite-element simulations used two-dimensional
elasticity with a plane-stress assumption, thereby simulat-
ing a free surface.
Results and discussion
Effect of SPO and LPO
The influence of SPO and LPO are illustrated for a calcitic
marble with a nominal grain size of 1 mm and a grain-
boundary toughness of 40 % of the single crystal toughness.
Nonetheless, the results are general for all compositions and
grain sizes. The results for this configuration are averaged
over three replications of orientations (i.e., the ODFs) for the
LPO distribution. For the equiaxed microstructure with a
random ODF (M = 1) the onset of microcracking on heating
occurs at an average temperature change of DT =
15 ± 2 C. For the mixed grain microstructure micro-
cracking commences at DT = 16 ± 1 C, while for the
elongated microstructure microcracking commences at
DT = 20 ± 2 C. The elastic strain energy density and the
maximum principal stress were monitored during heating
and cooling for temperature step changes of 1 C for both
cracked and uncracked samples. The uncracked simulations
are those for which the microcracking algorithm was not
activated. The thermoelastic response values are plotted for
the temperature interval from 10 to 50 C during heating and
from -10 to -50 C during cooling (see Fig. 6). The strain
energy density and the maximum principal stress for the
cracked state show a strong dependence on SPO for random
crystallographic texture.
Microcracking in simulations with a highly textured
LPO (M = 20) commences at a larger temperature incre-
ment from the unstressed state compared to the simulations
with no texture, i.e., a random ODFs (M = 1). The equi-
axed microstructure begins to microcrack at average of
DT = 18 ± 1 C, while the microstructure with mixed-
shape fabric starts to crack at DT = 18 ± 4 C, and the
elongated microstructure commences to crack at an even
higher temperature differential of DT = 23 ± 4 C. The
strain energy density and maximum principal stress for the
fabric parameters are plotted in Fig. 7. A significant
influence of SPO on the response of the cracked micro-
structures is observed upon heating; upon cooling the
responses are similar.
The decrement in elastic strain energy density on mi-
crocracking (i.e., the difference between the strain energy
density of the uncracked state and the cracked state) is the
largest in the equiaxed microstructure with random texture
(see Fig. 8). Since elastic strain energy is converted to
fracture surface energy on microcracking, this result would
suggest more microcracking in the equiaxed microstructure
with random texture. To verify this conjecture, the aver-
aged percentage of microcracked elements from three
replications of the ODF are shown in Fig. 9 for a temper-
ature change of 50 C in both heating and cooling. The
greatest extent of microcracking is observed in the equi-
axed microstructure with random texture upon cooling (i.e.,
9.1 ± 0.3 % of the total area at DT = -50 C). This is
closely followed by the same microstructure upon heating
(i.e., 7.8 ± 0.2 % at DT = ?50 C). In the mixed grain-
type microstructure only 7.2 ± 0.5 % of whole area is
microcracked upon cooling and 5.9 ± 0.3 % upon heating.
Even fewer microcracks are observed in the elongated
microstructure: 6.8 ± 1.3 % and 4.0 ± 0.6 % upon cool-
ing and heating, respectively. In all microstructures more
microcracking is observed upon cooling, than upon heat-
ing. The asymmetry is shown in Fig. 10 for an elongated-
grain microstructure with random texture. An apparent
reason for this asymmetry is not obvious.
Table 1 The single-crystal elastic constants, Cij, in GPa (Bass 1995) and the crystalline coefficients of thermal expansion, aij, in 10
-6 C-1 for
calcite (Kleber 1959) and dolomite (Reeder and Markgraf 1986)
Material C11 C12 C13 C14 C15 C33 C44 a11 a33
Calcite 144.0 53.9 51.1 -20.5 0 84.0 33.5 -6.0 26.0
Dolomite 205 70 57.4 -19.5 13.7 113.0 39.8 6.0 26.0
1268 Environ Earth Sci (2013) 69:1263–1279
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When the crystallographic texture is 20 times random,
the least amount of microcracking is observed in the
elongated grain-type microstructure upon heating. Only
0.3 ± 0.1 % of the total area microcracks for a temperature
change of DT = ?50 C. In contrast, cooling the same
microstructure results in 2.8 ± 0.4 % of the total area
microcracking for a temperature change of DT = -50 C.
In the mixed grain-type microstructure the amount of mi-
crocracking is 2.2 ± 0.4 % and 3.4 ± 0.3 % of the total
area upon heating and cooling, respectively. For the highly
textured material the most amount of microcracking is
observed in the equiaxed microstructure with 3.8 ± 0.1 %
of whole area upon heating and 4.0 ± 0.2 % upon cooling
(See Fig. 9). The influence of SPO on the degree of
microcracking is quite apparent upon heating: there is a
sevenfold increase in microcracking when changing from
the elongated grain-type to the mixed grain-type micro-
structure. Upon cooling, the amount of microcracking is
greater, but does not increase as significantly with
decreasing SPO (only about 20 % for each grain-type
change).
The amount of stored elastic strain energy of the
uncracked state, as measured by the strain energy density,
decreased approximately twofold with increasing LPO.
Fig. 6 Representative responses (a single replication of a random
ODF) of the elastic strain energy density and the maximum principal
stress for microcracked (red) and unmicrocracked (black) idealized
marble microstructures versus the temperature change from the
unstressed state. Results are shown for three idealized calcitic
microstructures (equiaxed grains, a mixture of equiaxed and elon-
gated grains, and elongated grains) with a nominal grain size of 1 mm
and random texture, denoted as M01
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However, the amount of stored strain energy of the cracked
state decreased only slightly with increasing LPO (Figs. 6,
7). Accordingly, the decrement in elastic strain energy
(Fig. 8) and the percentage of microcracked area (Fig. 9)
both decrease approximately by 60 % with increasing
LPO. Figure 11 provides visualization of the onset and
degree of microcracking and of the influence of SPO
and LPO.
In the present simulations initiation of microcracks was
generally observed to occur at grain triple junctions and
along single grain-boundary facets or partial facets.
Subsequent microcracking was mixed mode with a com-
bination of intergranular and transgranular fracture.
Spatial distribution of microcracks
To investigate the relation between maximum principal
stress and microcrack initiation and propagation, high-
stress regions in an uncracked material were compared to
the microcracks in the cracked material. Microstructural
response maps show the spatial dependence of micro-
cracking and maximum principal stress for an equiaxed
Fig. 7 Representative responses (a single replication of a textured
ODF) of the elastic strain energy density and the maximum principal
stress for microcracked (red) and unmicrocracked (black) idealized
marble microstructures versus the temperature change from the
unstressed state. Results are shown for three idealized calcitic
microstructures (equiaxed grains, mixture of equiaxed and elongated
grains, and elongated grains) with a nominal grain size of 1 mm and
crystallographic texture that is 20 times random texture (denoted as
M20)
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microstructure with a random ODF (Fig. 12) and for a
mixed grain microstructure with a high degree of LPO
texture (M = 20) (Fig. 13). Microstructural response maps
are shown for DT = 50 C and DT = -50 C in both
uncracked and cracked materials. As was pointed out by
Weiss et al. (2002) cracks that occurred in microstructure
upon heating do not form upon cooling and vice versa.
Regions with high maximum principal stress in the
uncracked state correspond to microcracks in the cracked
material. Most of these regions in cracked material have
high maximum principal stress as well but the values are
smaller than those in the uncracked state. For the uncracked
state in a mixed grain microstructure with M = 20 texture
there are fewer regions with high maximum principal stress
than in the equiaxed-grain microstructure with random tex-
ture, i.e., the propensity for microcracking is less.
Fig. 8 The decrement in the elastic strain energy density between the
unmicrocracked and the microcracked states of idealized calcitic
marble microstructures with a nominal grain size of 1 mm. Results
are shown for random crystallographic texture (denoted M01) and a
texture that is 20 times random (denoted as M20)
Fig. 9 Averaged percentage of microcracked elements from three
replications of the ODF for the three idealized calcitic marble
microstructures with a nominal grain size of 1 mm and random
texture (denoted as M01) and texture that is 20 times random (denoted
as M20). Results are for a temperature change of ?50 C (denoted as
heating) and -50 C (denoted as cooling). The error bars represent
one standard deviation
Environ Earth Sci (2013) 69:1263–1279 1271
123
Effect of grain size and marble composition
Extensive microcracking is expected to occur in coarse
grain microstructures composed of grains that have a high
thermal expansion anisotropy (Cleveland and Bradt 1978).
Indeed, given a temperature change, one expects the
amount of microcracking to diminish with grain size until a
critical grain size is reached, below which microcracking
does not occur in finer grain materials. For a temperature
change of DT = 50 C the critical grain size for calcite is
approximately 45 lm and that for dolomite is approxi-
mately 80 lm. Thus, in all the grain sizes considered here
microcracking is expected to occur for temperature chan-
ges of ±50 C. However, decreasing amounts of micro-
cracking are expected as the grain size decreases from
2 mm to 1 mm to 200 lm.
Accordingly, simulations were performed for calcitic
marbles with nominal grain sizes of 200 lm and 1 mm,
and for dolomitic marbles with nominal grain sizes of 1 and
2 mm. The percentages of microcracked area for the dif-
ferent grain sizes and marble compositions are presented in
Table 2. For the 1 mm grain-size calcitic marble the results
are averaged over three replications of the ODF to give a
mean and standard deviation. For all others fabric config-
urations the results are only for one replication of the ODF,
but one expects similar variability as that for the 1 mm
grain-size calcite.
As expected, the percentage of microcracked area for a
total temperature change of DT = 50 C is significantly
reduced for the 200 lm nominal grain-size calcite material
in comparison to the 1 mm grain-size material. In the
equiaxed-grain material with random crystallographic
texture the microcrack percentage is only 0.5 % of whole
area upon heating compared to 7.8 % for the 1 mm grain-
size material. Upon a negative temperature change (cool-
ing) the microcrack percentage is dimensioned even more
from 9.1 to 0.4 %. While the amount of microcracking for
a highly textured material (M = 20) is less than that for a
material with random texture (M = 1), and is not as
asymmetric in heating versus cooling, the reduction in the
amount of microcracking with grain size is more pro-
nounced: approximately one fortieth when reducing the
grain size from 1 mm to 200 lm. The mixed grain-shape
materials exhibit an even greater decrease in the percentage
Fig. 10 Simulations of
microcracking in an elongated-
grain calcite marble with
random texture and a nominal
grain size of 1 mm: a upon
heating by a temperature
differential ?50 C; b upon
cooling by a temperature
differential -50 C
Fig. 11 Representative responses (a single replication of each ODF)
of the percentage of microcracked elements for the three idealized
calcitic marble microstructures with a nominal grain size of 1 mm
upon heating and cooling. Random texture is denoted as M01; 20
times random texture is denoted as M20
1272 Environ Earth Sci (2013) 69:1263–1279
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Fig. 12 A cropped selection (170 9 170 pixels) from a microstruc-
tural response map showing the spatial dependence of the maximum
principal stress for an uncracked and cracked equiaxed-grain calcitic
marble microstructure with a nominal grain size of 1 mm and a
random ODF for a temperature changes of DT = -50 C and
DT = ?50 C
Fig. 13 A cropped selection (170 9 170 pixels) from a microstruc-
tural response map showing the spatial dependence of the maximum
principal stress for an uncracked and cracked mixed-grain calcitic
marble microstructure with a nominal grain size of 1 mm and a highly
textured ODF (M = 20) for a temperature changes of DT = -50 C
and DT = ?50 C
Environ Earth Sci (2013) 69:1263–1279 1273
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of microcrack area with grain-size reduction (see Table 2).
Moreover, the amount of reduction is strongly dependent
on the LPO. The 200 lm grain size, elongated grain-shape
microstructure showed the most significant reduction in
microcracking compared to the 1 mm grain-size material.
Almost no microcracking was observed for the 50 C
temperature change. Furthermore, the reduction in the
amount of microcracking for these fabric conditions was
strongly dependent on the LPO. For the 200 lm elongated-
grain microstructure the percentage of microcracked area
was 0.007 % upon heating and 0.03 % upon cooling when
the texture is random; and basically 0 % upon heating and
0.1 % upon cooling when there is high LPO texture.
The percentage of microcracked area is also reduced for
the 1 mm nominal grain-size dolomite material compared to
the 2 mm grain-size material (see Table 2). Furthermore, the
1 mm grain-size dolomite marble was observed to have less
propensity for microcracking than the 1 mm grain-size cal-
cite marble, i.e., microcracking begins at higher temperature
differential and the percentage of cracked area is less (see
Table 2). For instance, upon heating an equiaxed micro-
structure with random texture (M = 1) the onset tempera-
ture for thermal microcracking occurs at 14 C for calcite
marble, while the dolomite marble does not begin to crack
until 19 C. While dolomite is stiffer than calcite, calcite has
a greater thermal expansion anisotropy resulting in a higher
microstructural maximum principal stresses. The percentage
of microcracked area in dolomitic marbles with a nominal
grain size of 2 mm is comparable with results for calcitic
marble with a nominal grain size of 1 mm.
The simulations also reveal a significant influence of the
SPO and LPO fabric parameters on the temperature change
Table 2 Percentage of microcracked area for three microstructures:
equiaxed grains, a mixture of equiaxed and elongated grains, and
elongated grains of calcite and dolomite marbles with random texture
(denoted as M01) and with strong texture (denoted as M20) after
heating and cooling in 1 C increments for a total temperature
differential of 50 C
Equiaxed grains Mixed grains Elongated grains
M01 (%) M20 (%) M01 (%) M20 (%) M01 (%) M20 (%)
200 lm grain-size calcite
Heating 0.5 0.1 0.3 0.004 0.007 0.00
Cooling 0.4 0.1 0.2 0.07 0.03 0.01
1 mm grain-size calcite
Heating 7.8 ± 0.2 3.8 ± 0.1 5.9 ± 0.3 2.2 ± 0.4 4.0 ± 0.6 0.3 ± 0.1
Cooling 9.1 ± 0.3 4.0 ± 0.2 7.2 ± 0.5 3.4 ± 0.3 6.8 ± 1.3 2.8 ± 0.4
1 mm grain-size dolomite
Heating 4.9 1.6 2.8 0.6 1.4 0.04
Cooling 5.1 2.2 3.5 1.4 2.5 1.3
2 mm grain-size dolomite
Heating 8.4 2.6 6.4 2.8 5.0 0.4
Cooling 10.8 4.1 8.4 3.8 7.7 3.1
Fig. 14 Simulations of
microcracking in dolomite
marble with a nominal grain
size of 1 mm upon heating by a
temperature differential
?50 C: a equiaxed
microstructure with random
texture and grain-boundary
toughness of 0.40 cxtal;
b equiaxed microstructure with
random texture with grain-
boundary toughness of
0.20 cxtal. The red ellipses
identify representative area of
comparison
1274 Environ Earth Sci (2013) 69:1263–1279
123
(DT) require for microcrack initiation. The equiaxed-grain
calcite microstructure with random texture and a nominal
grain size of 200 lm begins to crack upon heating at
DT = 31 C and upon cooling at DT = -37 C. In the
mixed grain-shape calcite with random texture micro-
cracking commences at DT = 34 C upon heating and at
DT = -34 C upon cooling. Microcracking in the elon-
gated-grain calcite commences at DT = 45 C and at
DT = -40 C upon heating and cooling, respectively. For
the highly textured calcite (M = 20) a larger temperature
differential is required for microcracking, namely,
DT = 38 C and at DT = -41 C in the equiaxed
microstructure, DT = 48 C and DT = -34 C in mixed-
shape microstructure, and DT = 50 C and DT = -44 C
in the elongated-shape microstructure. Indeed, for the
elongated-shape fabric microcracking was just initiated at
these temperatures, i.e., the first mutated elements in OOF
(the procedure in OOF1 for modifying element properties
to simulate fracture) occurred at these temperatures.
Effect of grain-boundary toughness
Simulations for dolomite marble with grain size 1 mm have
been performed. The fracture algorithm assumes that the
Fig. 15 Polycrystalline thermal expansion coefficients in the e1- and e2-directions (denoted as ax and ay, respectively) upon heating and cooling
by a total temperature differential 50 and -50 C for a calcitic marble with a nominal grain size of 1 mm. Results are shown for one replication
Environ Earth Sci (2013) 69:1263–1279 1275
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critical fracture energy of the grain boundary is different
from that of the grain interior. Simulations with two grain
boundaries toughness are presented, to study the transition
from the intergranular to the transgranular mode of crack
growth.
More microcracking was observed when the grain-
boundary toughness (cig) was 20 % of the grain fracture
surface energy (cxtal) than when it was 40 % of cxtal.
Additionally, the microstructure with cig = 0.2 cxtal com-
mences to crack at a smaller temperature differential.
Spatial maps of the extent of microcracking for a total
temperature change of ?50 C are shown in Fig. 14. When
the grain-boundary toughness is as weak as 20 % of that of
the grains, transgranular cracking is essentially reduced
completely. Microcracks initiate on single grain-boundary
facets or on partial grain-boundary facets, and then prop-
agate along the grain boundaries.
Coefficients of thermal expansion
The thermal expansion coefficients were computed for a
given total temperature change, i.e., for DT = 50 C and
DT = -50 C, and a given microstructure configuration,
i.e., for a calcitic and a dolomitic marble with a nominal
grain size of 1 mm and a grain-boundary toughness
0.40 cxtal. The thermal expansion coefficient in the
e1-direction, denoted as ax, is computed from the relative
displacement change of the right and left sides of the
microstructure. The thermal expansion coefficient in the
e2-direction direction, denoted as ay, is computed from
the relative displacement change of the top and bottom of
the microstructure. Figure 15 illustrates the influence of
LPO and SPO on ax and ay. Numerical results are presented
in Table 3 for the calcitic and dolomitic marbles upon
heating and cooling for a total temperature differential of
50 C. Results are also shown for the case when no
microcracking occurs. For these simulations fracture, or
microcracking, was inhibited by turning off the cracking
algorithm in OOF.
In microstructures with no microcracking the coeffi-
cients of thermal expansion (the natural polycrystalline
values) are constant during heating and cooling, and the
same for both conditions. When microcracking is allowed
the coefficients of thermal expansion in the cracked
microstructure are greater than the natural polycrystalline
values during heating and less than the natural poly-
crystalline values during cooling (see Fig. 15 and
Table 3). This result is indicative of the extra expansion
of the microstructure upon microcracking. Microcracks
expand the microstructure. So, on heating this additional
extension adds to the thermal expansion of the micro-
structure, thereby giving a larger result. On cooling, this
additional expansion due to the microcracks subtracts
from the thermal contraction of microstructure, thereby
giving a diminished result. With increasing SPO (i.e.,
equiaxed grains to mixed grain shapes to elongated
grains) and increasing LPO (i.e., random crystallographic
texture, M = 1, to highly textured, M = 20), micro-
cracking becomes more spatially anisotropic. When this
happens, the difference between ax and ay, i.e., (ax - ay),
increases (see Fig. 15). Numerical results in Table 3 for
both calcitic and dolomitic materials show a strong
dependence on the degree of LPO, and almost no
dependence on SPO.
Table 3 The coefficients of thermal expansion in units of 10-6 K-1
in the e1 and e2 directions, are denoted as ax and ay, respectively,
upon heating and cooling by a temperature differential 50 and -50 C
for microstructure allowing microcracking and for microstructure
with no microcracking
Equiaxed Mixed Elongated
1 mm grain size calcite
M01
Heating ax = 9.84
ay = 9.27
ax = 10.2
ay = 8.48
ax = 9.59
ay = 7.28
Cooling ax = -0.64
ay = 0.04
ax = -1.00
ay = 0.38
ax = -0.11
ay = 0.80
No cracked ax = 4.38
ay = 4.26
ax = 4.81
ay = 3.96
ax = 5.63
ay = 3.85
M20
Heating ax = -0.005
ay = -0.301
ax = -0.488
ay = -1.043
ax = -1.580
ay = -2.529
Cooling ax = -3.838
ay = -3.940
ax = -3.290
ay = -3.622
ax = -2.512
ay = -3.553
No cracked ax = -2.221
ay = -2.370
ax = -1.734
ay = -2.350
ax = -1.648
ay = -2.597
1 mm grain size dolomite
M01
Heating ax = 14.9
ay = 14.4
ax = 14.5
ay = 10.9
ax = 14.1
ay = 12.7
Cooling ax = 10.5
ay = 10.9
ax = 11.1
ay = 10.8
ax = 12.1
ay = 11.2
No cracked ax = 12.4
ay = 12.3
ax = 12.7
ay = 12.1
ax = 13.3
ay = 12
M20
Heating ax = 8.9
ay = 8.63
ax = 8.8
ay = 8.2
ax = 8.76
ay = 7.99
Cooling ax = 7.66
ay = 7.55
ax = 8
ay = 7.8
ax = 8.54
ay = 7.67
No cracked ax = 8.3
ay = 8.18
ax = 8.6
ay = 8.2
ax = 8.75
ay = 7.99
Results are presented for calcite and dolomite marble with a nominal
grain size of 1 mm
1276 Environ Earth Sci (2013) 69:1263–1279
123
Fig. 16 Idealized illustration of results for calcitic and dolomitic
marbles with a grain size of 1 mm for a temperature change of
DT = -50 C. Results are shown for three cropped selections of
extremes of microstructures (equiaxed grains, a mixture of equiaxed
and elongated grains, and elongated grains) with random texture
(different colors correspond to different random orientations) and
strong texture [similar colors correspond to similar crystalline
orientations (i.e., a textured ODF)]. With increasing the degree of
texture and shape preferred orientation microcracking (red curves) is
decreasing
Environ Earth Sci (2013) 69:1263–1279 1277
123
Summary and conclusions
The thermoelastic response of marble and concomitant
degradation by microcracking depend on many fabric
parameters. The present study revealed certain combination
of shape preferred orientation (SPO), lattice preferred ori-
entation (LPO), grain size, grain-boundary toughness, and
marble composition that have significant influence on the
thermomechanical and degradation behavior of marble.
The elastic strain energy density and the maximum
principal stress that result from the thermal expansion
anisotropy of the marble crystalline grains and their spatial
distribution in the microstructure are excellent indicators of
microcracking. Regions in the microstructure with high
values of these two microstructural properties in the
uncracked state are expected to indicate regions with a
propensity for microcracking (Shushakova et al. 2011). It
was observed that regions with high maximum principal
stress in the uncracked state correspond to microcracks in
the cracked material. The onset and degree of micro-
cracking strongly depend on the fabric parameters. With
decreasing grain size and increasing LPO and SPO
microcracking is less prominent and occurs at a larger
temperature differential. Microcracking upon heating is
predominantly less than that upon cooling (see Fig. 9).
For the case of the mixed-grain and the elongated-grain
microstructures with strong crystallographic texture, this
asymmetry is significant. The percentage of microcracked
area upon heating is one-tenth of that upon cooling.
Marble composition plays an important role in the
degree of microcrack degradation. Microcracking in a
dolomitic marble commences at higher temperature dif-
ferential and exhibits less of a tendency to microcrack than
for a calcitic marble with the same microstructure and
texture. Thus, finite-element simulations indicate that
dolomitic marbles are more resistant against thermal deg-
radation. While dolomite is stiffer than calcite, calcite has a
greater thermal expansion anisotropy, resulting in higher
maximum principal stresses. In agreement, Zeisig et al.
(2002) showed experimentally that dolomitic marbles do
not have residual strains after thermal treatment. To illus-
trate the influence of fabric parameters such as marble
composition, SPO and LPO on microcracking Fig. 16 is
created.
Another fabric parameter that affects microcracking is
the grain-boundary toughness (cig). In the simulations the
grain-boundary toughness was chosen to be 40 and 20 % of
the grain fracture surface energy (cxtal) to elucidate the
fracture behavior of both transgranular and intergranular
crack-growth modes and to bracket the value of the grain-
boundary fracture toughness that delineates the transition
from transgranular fracture to intergranular fracture. For
the case of cig = 0.4 cxtal microcracking initiated as a
grain-boundary microcrack either along a single grain-
boundary facet or partially along such a facet. Subsequent
microcrack extension occurred predominantly along grain
boundaries, i.e., intergranular fracture, but occasional
excursions through some grains were observed, i.e., intra-
granular cracks. When the grain-boundary toughness
decreased to 20 % of cxtal, the amount of intragranular
cracks was greatly reduced, or vanished all together
(e.g., see Fig. 14).
Finally, microstructure-based finite-element modeling
provides excellent insight and elucidation of influences of
rock fabric and crystal texture on the thermoelastic
behavior and microcracking response of marbles.
Acknowledgments The authors gratefully acknowledge David M.
Saylor for generating the artificial microstructures used in this study
with the Microstructure Builder program, which he was developing
in collaboration with Carnegie Mellon University and Alcoa
Technical Center. Thomas Weiss is gratefully acknowledged for
providing the ESEM figures and B.Middendorf as well as B.Grelk
for Fig. 1 and Fig. 2b. Financial support for E.R. Fuller, Jr. at
Universita¨t Go¨ttingen was provided by the Deutsche Forschungs-
gemeinschaft grant: SI 438/39-1, and is gratefully acknowledged.
Victoria Shushakova gratefully acknowledges a long-term DAAD
fellowship grant.
Open Access This article is distributed under the terms of the
Creative Commons Attribution License which permits any use, dis-
tribution, and reproduction in any medium, provided the original
author(s) and the source are credited.
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