SPECIAL ISSUE
Influence of shape fabric and crystal texture on marble
degradation phenomena: simulations
Victoria Shushakova • Edwin R. Fuller Jr •
Siegfried Siegesmund
Received: 19 August 2010 / Accepted: 11 September 2010 / Published online: 28 September 2010
 The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract Microstructure-based finite element simula-
tions were used to study the influence of grain shape fabric
and crystal texture on thermoelastic responses related to
marble degradation phenomena. Calcite was used as an
illustrative example for studying extremes of shape pre-
ferred orientation (SPO) in shape fabric and lattice pre-
ferred orientation (LPO) in crystal texture. Three SPOs
were analyzed: equiaxed grains, elongated grains, and a
mixture of equiaxed and elongated grains. Three LPOs
were considered: a random orientation distribution function
and two degrees of strong directional crystal texture.
Finally, the correlation between the direction of the LPO
with respect to that of the SPO was examined. Results
show that certain combinations of SPO, LPO, and their
directional relationship have significant influence on the
thermomechanical behavior of marble. For instance, while
there is no major dependence of the elastic strain energy
density and the maximum principal stress on SPO for
randomly textured microstructures, there is a strong syn-
ergy between LPO and its directional relationship with
respect to the SPO direction. Microcracking precursors,
elastic strain energy density, and maximum principal
stress, decrease when the crystalline c-axes have fiber
texture perpendicular to the SPO direction, but increase
significantly when the c-axes have fiber texture parallel to
the SPO direction. Moreover, the microstructural variabil-
ity increases dramatically for these latter configurations. In
general, the influence of LPO was as expected, namely, the
strain energy density and the maximum principal stress
decreased with more crystal texture, apart from for the
exception noted above. Spatial variations of these precur-
sors indicated regions in the microstructure with a pro-
pensity for microcracking. Unexpectedly, important
variables were the microstructural standard deviations of
the spatial distributions of the microcracking indicators.
These microstructural standard deviations were as large as
or larger than the variables themselves. The elastic misfit-
strain contributions to the coefficients of thermal expansion
were also calculated, but their dependence was as expected.
Keywords Shape fabric  Shape preferred orientation 
Crystal texture  Lattice preferred orientation  Marble 
Calcite  Finite element simulations  Thermal expansion
anisotropy  Elastic strain energy density  Maximum
principal stress  Orientation distribution function 
Coefficient of thermal expansion
Introduction
Marbles have been considered to be among the most
important building materials since ancient times. Regarded
as special stones, they are captivating because of their
pureness, formability, and translucence. However, degra-
dation of sculptures, architectural heritage, and fac¸ade
stone made from marbles is problematic. Such deteriora-
tion of building stones depends mainly on climate. Many
experimental studies have shown that temperature changes,
both increases and decreases, induce significant deteriora-
tion. Even when the temperature changes are not particu-
larly large, repeated heating and cooling of stone will
eventually lead to deterioration over time, e.g., Kessler
(1919), Battaglia et al. (1993), Winkler (1994), Sieges-
mund et al. (2000), Zeisig et al. (2002). This deterioration
is most spectacular for marbles, but is also observed for
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 (2011) 63:1587–1601
DOI 10.1007/s12665-010-0744-7
other rocks, like granite, Nagelfluh, and limestone, e.g.,
Siegesmund and Du¨rrast (2011).
The rock-forming minerals in marble, like calcite and
dolomite, have large anisotropy in their coefficients of
thermal expansion. Hence, apart from the expansion or
contraction that results from heating or cooling, residual
stresses will develop in the stone due to the thermal
expansion anisotropy between the constitutive crystalline
grains. These stresses can result in thermally induced mi-
crocracking, and an additional concomitant expansion.
This behavior is exacerbated by moisture due to moisture-
assisted subcritical crack growth (Henry et al. 1978;
Atkinson 1984), and hence degradation can accumulate
with cycling and over time, resulting in significant time-
dependent deterioration (Koch and Siegesmund 2004).
For some marbles, the degradation process is expressed
as a penetrative granular disintegration. This phenomenon
is typical for Carrara marbles, although it is not restricted
to only this marble type. At the sculpture surfaces a pro-
gressive fabric decohesion leads to a sugar-like crumbling
of the isolated crystalline grains of calcite, resulting in a
progressive deterioration of the stone. An example of this
degradation phenomenon is shown in Fig. 1a. The
Madonna and Child statue of Carrara marble is generally
well preserved due to a thin, hardened outer layer. How-
ever, the interior rock fabric has undergone a progressive
decohesion along the grain boundaries, resulting in loss of
the facial features.
A more spectacular deterioration phenomenon occurs
for marble slabs, namely their bowing behavior. Due to
progress in natural stone machining, unprecedented crea-
tive options are now available for economical cladding
with natural stone slabs. This has led to a renaissance in the
use of natural stone as a modern building material. Despite
the aesthetic gains, there are structural and aesthetic issues.
Exposed to the weather, stone claddings may be disag-
gregated in their structure, and thus irreversibly deformed.
The bowing of marble fac¸ade slabs, or gravestones (see
Fig. 1b), with thicknesses of typically 30 to 40 mm, but
occasionally up to 90 mm, have been reported (Grimm
1999; Siegesmund et al. 2008). Both panel thickness and
temperature influence the development of bowing (Koch
and Siegesmund 2002, 2004; Grelk et al. 2004). Moreover,
both convex and concave deformation can occur after only
a few years of exposure. This kind of deformation was
already described in the literature as early as 1919 by
Kessler (1919), and more recently by Sage (1988) and
Logan et al. (1993). They found that repeated heating of
marble can cause a permanent expansion via damage to the
microstructure, and thereby to the cladding. The process
behind this phenomena is still under debate. Microstruc-
ture-based finite element simulations were used by Weiss
et al. (2002, 2003, 2004), Saylor et al. (2007), and Wanner
et al. (2010) to provide excellent insight into the influence
of crystal texture (lattice preferred orientation) and grain-
neighbor misorientation configurations on the microstruc-
tural strain energy and stresses affecting thermal degrada-
tion. The basic observation is that the rock fabric has a
significant influence on the residual strains that occur
during thermal excursions, thus on thermal degradation.
Fig. 1 Two examples of marble
degradation phenomena.
a Madonna and Child statue of
Carrara marble on a tombstone
in the historic Munich cemetery.
The statue is generally well
preserved due to a thin,
hardened outer layer. However,
the interior rock fabric exhibits
a progressive granular
decohesion along the grain
boundaries, the so-called
sugaring phenomenon. b A
warped gravestone of Carrara
marble in the Montmartre
Cemetery, Paris, France
1588 Environ Earth Sci (2011) 63:1587–1601
123
Moreover, the thermal expansion behavior of marbles was
modeled with a good coincidence to real experiments.
Systematic experimental studies by Zeisig et al. (2002) and
Siegesmund et al. (2008) showed that grain size, lattice
preferred orientation (LPO), as well as the shape preferred
orientation (SPO) can cause a different amount of residual
strain. They pointed out that different calcite marbles will
have different coefficients of thermal expansion because
the size and orientation of the calcite crystalline grains as
well as their shape will be different.
The objective of this paper is to elucidate the synergistic
influence of shape fabric (e.g., SPO) and crystal texture
(e.g., LPO) on factors, such as the stored elastic strain
energy density and microstructural stresses, that influence
marble degradation. Calcite is used as an illustrative
example, but results are expected to be general for myriad
marble microstructures, as the thermophysical properties of
various marbles do not differ that much. Microstructure-
based finite-element simulations are used to explore a wide
range of SPO and LPO. We examine the extremes of shape
fabric: an equiaxed and an elongated microstructure with
an intermediate case that is a mixture of the two extremes,
as illustrated in Fig. 2. We also examine extremes of
crystal texture from a random orientation distribution
function (ODF) to ODFs that are crystallographically tex-
tured at 20 and 40 times a random ODF, also as illustrated
in Fig. 2. Additionally, the directionality of the LPO with
respect to the SPO is examined and is found to have a
significant influence.
Microstructural simulations
Simulation design
The simulation design matrix has 3 9 3 9 3 or 27 vari-
ables. The three SPOs are an equiaxed microstructure, a
mixed microstructure, and an elongated microstructure (see
Fig. 3). The three LPOs are a random ODF, an ODF with
fiber-texture that is 20 times random, and an ODF with
fiber-texture that is 40 times random. The last three vari-
ables are the directionality of the LPO fiber texture with
respect to the SPO. These directional conditions are crystal
fiber texture perpendicular to the SPO, but in the plane of
the microstructure; crystal fiber texture parallel to the SPO;
and crystal fiber texture perpendicular to the SPO, but out of
the plane of the microstructure. For the random ODF these
three directional conditions are equivalent and are not dis-
tinct. Accordingly, there are only 21 unique configurations
specified by shape fabric, crystal texture, and directionality
of the LPO with respect to the SPO. For each of these 21
cases five different, independent sets of orientations, or
replications, were generated. Thus, 105 microstructural
configurations served as the basis for the finite-element
simulations.
Shape fabric
The two-dimensional simulated rock microstructures were
generated by a nucleation and growth algorithm, which is
described in detail elsewhere (Ito and Fuller 1993;
Miodownik et al. 1999; Saylor et al. 2004, 2007). Circular
or elliptical seeds were used to give the extremes in shape
fabric. For circular (isotropic) seeds the simulated equiaxed
microstructure is a Voronoi tessellation (see Fig. 3a). For
elongated elliptical seeds the simulated microstructure is
elongated. By giving directionality to the seeds, in this case
along the y-axis, the textured shape fabric (or SPO) shown
in Fig. 3c is obtained. Using a mixture of the two types of
seeds, the mixed shape fabric shown in Fig. 3b is obtained.
The microstructural images have a resolution of
1,000 9 1,000 pixels. The number of grains in the equi-
axed, mixed, and elongated microstructures, respectively,
are 382 grains, 347 grains, and 312 grains. Accordingly,
the average grain size increases with shape fabric from
2,617.8 pixels per grain for the equiaxed microstructure to
2,881.8 pixels per grain for the mixed microstructure to
3,205.1 pixels per grain for the elongated microstructure.
The coordinate system, shown in Fig. 3, is used to
describe the results: the y direction is parallel to the SPO;
the x direction is perpendicular to the SPO and in the plane
of the microstructure; and the z direction is perpendicular
to the SPO and out of the plane of the microstructure.
Crystal texture
Orientation distribution functions (ODFs) with LPOs were
generated via the March Dollase fiber-texture distribution
(Dollase 1986; Blendell et al. 2004). Given a texture
direction and the crystallographic axes to be textured, in
this case the crystallographic c-axes, the cumulative frac-
tion of c-axis orientations within a cone of half angle h
about the texture direction is given by:
PðM; hÞ ¼ 1  cosðhÞ =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
cos2ðhÞ þ M sin2ðhÞ
q
ð1Þ
where M is a parameter characterizing the degree of
texture. The probability distribution function is given by
oPðM; hÞ=oh ¼ f ðM; hÞsin hð Þ, where
f ðM; hÞ ¼ M=½cos2ðhÞ þ M sin2ðhÞ3=2 ð2Þ
gives the multiple of a random distribution (MRD) at that
value of h.
If M = 1, f(M = 1,h) = 1, and the distribution is ran-
dom. If M [ 1, one has fiber texture in a cone of half angle
h (the c-axis fiber types considered here). The maximum
Environ Earth Sci (2011) 63:1587–1601 1589
123
MRD occurs at h = 0, and is equal to M. Perfect fiber
texture is defined as texture with axial symmetry. It can be
envisioned as a single crystal rotating around a specific
axis. In the case of c-axis perfect fiber texture, the rotation
axis is the c-axis. There is a single c-axis maximum, the
a-axes are distributed on a great circle, and the normals of
other crystallographic planes are distributed on small
circles. If M \ 1, a case not considered here, one has girdle
texture outside a cone of half angle h. The maximum MRD
occurs at h = p/2, and is equal to 1/HM. Such distributions
with M \ 1 could be used to describe the a-axis fiber types.
Three ODFs (or three LPOs) were considered: a random
ODF (i.e., M = 1); an ODF with fiber-texture that is 20
times random (i.e., M = 20); and an ODF with fiber-
Fig. 2 A schematic illustration of the extremes of shape fabric and
crystalline texture. From left to right are shown two extremes of shape
fabric and a mixture of the extremes: a, d an equiaxed microstructure;
c, f an elongated grain shape; b, e a mixed microstructure with shape
fabric from the two extremes. From top to bottom are shown two
extremes of crystallographic texture: a–c random crystallographic
texture; and d–f extreme crystallographic texture. Also examined, but
not shown is the directionality of the crystallographic texture with
respect to the shape fabric, i.e., either parallel or perpendicular to the
shape fabric
Fig. 3 The three microstructures used in the simulations to vary
shape fabric: a an equiaxed microstructure; b a mixed microstructure
with shape fabric from the two extremes; and c an elongated grain
shape microstructure. The number of grains in each microstructure,
respectively, are 382 grains, 347 grains, and 312 grains. The
coordinate systems used to describe the results is also indicated in
the figure
1590 Environ Earth Sci (2011) 63:1587–1601
123
texture that is 40 times random (i.e., M = 40). For brevity
in labeling the simulation results, the three LPOs are
denoted as M1, M20, and M40, respectively. For all cases
the half-angle h of the cone containing the crystal c-axis
was selected from the March–Dollase distribution descri-
bed by Eq. (1). Using a random number between 0 and 1
for the cumulative probability P, the polar angle of the
orientation is given by:
h ¼ arccos
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M ð1  PÞ2
1 þ ðM  1Þð1  PÞ2
s
 !
ð3Þ
The two Azimuthal angles describing the crystal
orientation, u and x, were selected randomly from the
interval 0 to 2p.
Representative pole figures for the three ODFs consid-
ered here are shown in Fig. 4. These pole figures are for
one of the five replications generated for the equiaxed
microstructure, but are typical of all 105 pole figures. The
fiber texture is along the North–South direction in these
figures, but variants are considered where texture direction
is rotated with respect to the SPO. The pole figures 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. The distributions
become more textured as the maximum MRD in the
March–Dollase function (i.e., M) varies from 1 (untex-
tured) to 40 (highly textured).
Crystal texture direction
Both the LPO and the SPO have directionality. For textured
microstructures there are three possible relations between
them. Since here the SPO is always aligned along the
y-axis, these three relations are specified by the orientation
of the LPO. The designations x, y, and z, are used to cor-
respond to the c-axes having fiber texture along x-axis (in
plane and perpendicular to the shape fabric), y-axis (par-
allel to the shape fabric), and z-axis (out of plane and
perpendicular to the shape fabric), respectively.
Microstructure-based finite-element approach
The microstructure-based finite-element approach used
here is based on the Object-Oriented Finite Element pro-
gram (OOF) developed at the National Institute of Stan-
dards and Technology (Langer et al. 2001). The OOF
software is in the public domain. Executables, source code,
and manuals are available at: http://www.nist.gov/msel/
ctcms/oof/. The OOF1 software was used here.
The finite-element meshing procedure is described in
detail elsewhere (Langer et al. 2001; Weiss et al. 2002;
Chawla et al. 2003; Saylor et al. 2007; Wanner et al.
2010). Briefly, a uniform 200 9 200 mesh of 80,000 right
triangular elements was positioned on the 1,000 9 1,000
pixel microstructure. Then an adaptive meshing algorithm
was used to align the nodes of inhomogeneous elements
(i.e., those which overlapped two or more grains) with the
grain boundaries. The resulting finite-element meshes
consisted of 80,000 triangular elements, which have an
average area of 12.50 ± 3.80 pixels per element for the
equiaxed microstructure, 12.50 ± 3.74 pixels per element
for the mixed microstructure, and 12.50 ± 3.66 pixels per
element for the elongated microstructure. The element
size distribution was kept relatively uniform so that
microstructural statistics over the microstructure are more
meaningful.
After the finite-element mesh was generated, thermo-
physical properties of calcite (trigonal symmetry) were
assigned to the elements. The single-crystal values for the
coefficients of thermal expansion (Kleber 1959) and the
Fig. 4 Representative pole figures showing the individual poles of
the crystallographic c-axis for each grain in the microstructure. In
a the orientations were chosen from a random ODF. In b the
orientations were chosen from a March-Dollase ODF with fiber
texture along the North and South pole that is 20 times random. In
c the orientations were chosen from a March–Dollase ODF with fiber
texture along the North and South pole that is 40 times random. For
both b and c the contour lines correspond to multiple random
distribution (MRD) from the March-Dollase distribution function of
16 (blue), 4 (green), 1 (orange) and 1/2 (red)
Environ Earth Sci (2011) 63:1587–1601 1591
123
elastic stiffness coefficients (Bass 1995) are given in
Tables 1 and 2, respectively. The crystallographic orien-
tations of each grain were selected from the appropriate
ODF, as described above, and were from three-dimensional
distributions.
Upon heating or cooling from a stress-free state, the
misorientation between neighboring grains and the asso-
ciated thermal expansion anisotropy result in thermal misfit
strains. These strains in turn give rise to internal residual
stresses, which are computed by the finite element method.
The temperature change used here was a temperature
increase of ?100C (i.e., heating). All of the finite-element
simulations used two-dimensional elasticity with a plane-
stress assumption, thereby simulating a free surface.
Effective polycrystalline in-plane coefficients of thermal
expansion coefficient were computed by measuring the
relative dimensional changes of the simulation micro-
structure and dividing by the temperature change. These
effective values represent only the contribution to the
thermal expansion (or contraction) from the elastic thermal
misfit strains. If microcracking occurs, for example in a
real microstructure, or if it is included in the simulations,
an additional expansion will occur, which is independent of
whether the temperature change is heating or cooling. The
total thermal expansion (or contraction) is the sum of the
elastic thermal expansion (contraction) and the micro-
cracking expansion.
Results and discussion
Shape preferred orientation (SPO) and lattice preferred
orientation (LPO)
Myriad SPO and LPO are observed for calcite in limestone
and marble. Calcite is easily deformable at low tempera-
tures, as is known from natural examples and from labo-
ratory deformation studies (e.g., Leiss and Molli 2003).
SPO is common in calcite and mainly results via plastic
deformation. Nonetheless, recrystallization and grain-
coarsening can reset the grain shapes to an equiaxed fabric
(Shelley 1993).
As is the case for most rock-forming minerals, LPO
for calcite strongly depends on the active slip systems,
and on the geometry and symmetry of the flow pattern.
This results in a LPO geometry that is similar to the
strain geometry (Wenk et al. 1987; Kern and Wenk 1983;
Shelley 1993; Passchier and Throuw 1996). For example,
in uniaxial compression, calcite develops a LPO with the
c-axes close to the shortening direction and at a high
angle to foliation, i.e., c-axes have a maximum around
the compression axis. In constriction a c-axes-girdle
develops perpendicular to an a-axis. Geometrically, the
fundamental LPO-types for calcite (i.e., perfect texture)
can be simply described by a rotating single crystal about
the c-axis or about one of the a-axes (c- and a-axis fiber-
type), or by any continuous sequence between these two
idealized end-members, e.g., Leiss and Ullemeyer (1999)
or Weiss et al. (1999).
Elastic strain energy density
The elastic strain energy density, U, is a key indicator of
potential microcracking sites in a microstructure, as it
provides the surface energy necessary to create the fracture
surfaces of the microstructural cracks. Accordingly, to
elucidate factors related to marble degradation from mi-
crocracking phenomena, the influences of shape fabric and
crystallographic texture on the elastic strain energy density
were studied by microstructural simulations. Variables
observed were (1) the average elastic strain energy density
for the microstructure and (2) the variation of the strain
energy density throughout the microstructure, as charac-
terized by the standard deviation of the strain energy
density distribution.
Elastic strain energy density: influence of SPO
The influence of shape fabric on the elastic strain energy
density, as characterized by the three SPO types (equiaxed,
mixed, and elongated), is minimal when the crystallo-
graphic texture is random (i.e., M = 1, or no LPO). See M1
in Fig. 5. The average strain energy density shows a small
decrease from 31.5 ± 0.7 kJ m-3 for the equiaxed micro-
structure to 29.8 ± 2.1 kJ m-3 for the mixed microstruc-
ture and to 28.3 ± 1.5 kJ m-3 for the elongated
microstructure. The variation represents the standard
deviation over the five simulation replications using five
different random ODFs.
The variation of the strain energy density within the
microstructure, microstructural standard deviation, simi-
larly shows little dependence on shape fabric for random
crystallographic texture. The microstructural standard
Table 1 The single-crystal coefficient of thermal expansion for cal-
cite in units of C-1 (Kleber 1959)
a11 = a22 a33
-6.0 9 10-6 26.0 9 10-6
Table 2 The single-crystal elastic stiffness coefficients for calcite in
units of GPa (Bass 1995)
C11 C12 C13 C14 C33 C44
144.0 53.9 51.1 -20.5 84.0 33.5
1592 Environ Earth Sci (2011) 63:1587–1601
123
deviation of the strain energy density distribution increases
from 29.7 ± 0.8 kJ m-3 for the equiaxed microstructure to
30.3 ± 2.0 kJ m-3 for the mixed microstructure and to
30.8 ± 1.1 kJ m-3 for the elongated microstructure.
Again, the variation is the standard deviation from the five
simulation replications of the random ODF.
Note that the variation of the strain energy density
within the microstructure is as large as the average strain
energy density. The frequency distributions of strain
energy density throughout the microstructure is essentially
a decreasing function with the largest fraction of strain
energy densities in the microstructure, as discretized by the
finite elements, lying at low values. The upper tail of the
frequency distribution of strain energy density is expected
to be characteristic of the propensity for microcracking.
However, a measure of this upper tail, i.e., the strain energy
density at 90% cumulative probability, essentially mimics
the average strain energy density, so it is not reported here
separately.
When the LPO has fiber texture that has a maximum
value of 20 times random, i.e., M = 20, the strain energy
density shows a significant dependence on the shape fabric.
Moreover, this dependence is strongly influenced by the
direction of the LPO with respect to the SPO. See M20 in
Fig. 5. For the equiaxed microstructure there is no SPO.
Thus, the strain energy density values should statistically
be the same whether the crystal texture is along the x- or
the y-directions. This is observed to be the case. However,
for the mixed and the elongated microstructures the strain
energy density value is significantly larger when the c-axis
crystal texture is aligned with the SPO, i.e., the y direction.
In contrast, when the c-axis crystal texture is perpendicular
to the SPO, i.e., the x direction, the strain energy density
value decreases with shape fabric. When the c-axis crystal
Fig. 5 Elastic strain energy
density and its microstructural
standard deviation (the square
root of its variance throughout
the microstructure) as a function
of SPO and LPO for a
temperature change of ?100C.
The different LPOs, or crystal
texture ODFs, are denoted as
M1, M20 and M40, respectively
corresponding to a March–
Dollase parameter M of 1, 20,
and 40 for the distribution from
which the orientations were
chosen. The errors bars show
the standard deviation from five
independent replications of the
ODF. Three directions for the
LPOs were observed: x, y and z,
corresponding to the c-axes
having fiber texture along x-, y-,
and z-axis, respectively
Environ Earth Sci (2011) 63:1587–1601 1593
123
texture is out of the plane of the simulation, i.e., the
z direction, the strain energy density values are smaller and
exhibit only a minor dependence.
However, the variation of the strain energy density, as
characterized by the microstructural standard deviation,
increases with increasing shape fabric for all three direc-
tions of the LPO (x, y and z) and is the largest for the
elongated microstructure. Furthermore, the microstructural
variation is substantially larger than the average value, and
the standard deviation of these standard deviation values
increases with increasing fabric shape. See Fig. 5.
When the LPO has fiber texture that has a maximum
value of 40 times random, i.e., M = 40, the strain energy
density exhibits a similar dependence with shape fabric and
LPO to that of M20, but slightly smaller. See M40 in
Fig. 5. However, the standard deviation of these standard
deviation values is larger than that of M20, i.e., there is
more variation in the five replications of the ODF.
Elastic strain energy density: influence of LPO
With one exception, as the crystal texture increases from
untextured or random (M = 1) to a high degree of fiber
texture (M = 40), the average strain energy density and its
microstructural variation (the microstructural standard
deviation) decrease. Thus, in most cases increased LPO or
crystal texture reduces the propensity for microstructural
microcracking.
The one exception is for an elongated microstructure
when the crystallographic c-axis texture is aligned parallel
with the SPO direction (i.e., the crystalline c-axes are
aligned along the shape fabric y direction). For this com-
bination of LPO and SPO, the average strain energy density
first increases with crystal texture from 28.3 ± 1.5 kJ m-3
for M = 1 to 30.8 ± 1.8 kJ m-3 for M = 20 and then
decrease with further crystal texture to 21.7 ± 5.3 kJ m-3
for M = 40. Similarly, the microstructural standard devi-
ation of the strain energy density distribution increases
with crystal texture from 30.8 ± 1.1 kJ m-3 for M = 1 to
41.4 ± 2.1 kJ m-3 for M = 20 and then decrease to
37.4 ± 4.5 kJ m-3 for M = 40. The reason for this phe-
nomenon is that several elongated grains per chance are
crystallographically misoriented with respect to the aver-
age LPO. Initially, the elastic strain energy from these
misaligned grains dominate the strain energy density dis-
tribution, giving rise both to more microstructural variation
and to a slight increase in the average value. With
increasing crystalline texture alignment, the effect dimin-
ishes. However, the phenomenon, which is described fur-
ther in the section on microstructural stress maps, is
probably unrealistic for two reasons. First, the crystal
texture is probably correlated with the shape texture, so
these misoriented grains are not highly probable. Second,
typical LPOs for calcite marbles are such that the c-axes
are aligned perpendicular to the SPO, i.e., the crystalline
c-axes are aligned along the shape fabric x or z directions
(e.g., Leiss and Ullemeyer 1999).
Maximum principal stress: influence of SPO
The dependence of the maximum principal stress and its
microstructural standard deviation on the SPO and the LPO
is essentially the same as that for the elastic strain energy
density. Figure 6 shows the average maximum principal
stress and the microstructural standard deviation for the
microstructure configurations investigated in this study.
The error bars represent the standard deviation for the five
replications of the ODF.
For the random ODFs (M1 in Fig. 6) there is slight, but
insignificant decrease in the maximum principal stress with
shape fabric: 34.7 ± 0.4 MPa for the equiaxed fabric;
33.1 ± 1.1 MPa for the mixed fabric; and 30.5 ± 0.9 MPa
for the elongated fabric. For the textured ODFs (M20 and
M40 in Fig. 6) the values for the x and y directions are
statistically equivalent for the equiaxed fabric, and
respectively, show decreases and increases with shape
fabric that are similar to those for the strain energy density.
The most significant variation with shape fabric is seen for
the M20 microstructural standard deviation when the
crystal texture is parallel to the shape texture (the y direc-
tion): 29.9 ± 1.5 MPa for the equiaxed fabric;
35.4 ± 2.1 MPa for the mixed fabric; and 47.2 ± 1.6 MPa
for the elongated fabric. As in the discussion for the strain
energy density, this phenomenon may be unrealistic. For
the highly textured ODFs (M40) all values are decreased
from the M20 values.
Maximum principal stress: microstructural maps
Spatial variations of the maximum principal stress are
shown in Fig. 7 for the three shape fabrics, the three degree
of crystallographic texture, and the three directions of LPO
with respect to SPO. The stress values are shown with a
thermal scale that ranges from black (0 MPa) to red-orange
(125 MPa) to white (C250 MPa). Each of the 27 micro-
structural maps shows the results for one of the five rep-
lications of the ODFs for that configuration. However, as
the results for the M = 1 rows are for a random ODF, the
maps in Fig. 7a, b, c for each shape fabric of this row are
statistically equivalent. Similarly, but now due to the
equiaxed fabric, the maps in Fig. 7a, b for the equiaxed
fabric column are statistically equivalent. Comparing these
equivalent maps, one can see the variations that arise from
the different ODFs and from the different equiaxed fabric
directions. These variations are reflected quantitatively by
the standard deviations of the maximum principal stress
1594 Environ Earth Sci (2011) 63:1587–1601
123
values (the error bars in Fig. 6). Similar variations occur
for the other replications of the ODFs for M = 20 and
M = 40 that are not reflected in Fig. 7. The statistical
variation between replications of the ODFs is not to be
confused with the microstructural standard deviation,
shown in Fig. 6 (and in Fig. 5 for the strain energy den-
sity). The microstructural standard deviation represents the
standard deviation of the maximum principal stress values
(or strain energy density values) within a single micro-
structural configuration.
For random crystal texture (M = 1) the overall range of
the maximum principal stress values does not change sig-
nificantly with SPO. However, the patterns in the stress
maps reflect the shape fabric. For the equiaxed fabric, and
to some extent for the mixed fabric, the stress patterns
show the network structure described by Wanner et al.
(2010). This structure of a high-stress network surrounding
low-stress regions (the darker regions in Fig. 7) develops
due to neighborhoods of grains having similar crystallo-
graphic orientations, thereby locally minimizing the ther-
mal expansion anisotropy misfit strains. This network
structure persists for microstructure with increasing in-
plane LPO (i.e., for M = 20 and M = 40, when the crystal
texture is along the x or the y direction, i.e., in-plane).
For increasing in-plane crystal texture (M = 20 and
M = 40) and SPO, the regions with relatively high maxi-
mum principal stress decrease when the crystal texture is
perpendicular to the SPO, i.e., in the x direction (see
Fig. 7a). This is the typical LPO for calcite marbles (e.g.,
Leiss and Ullemeyer 1999). Hence, the propensity for
microcracking should decrease for calcite marbles with
increasing LPO and SPO.
In contrast, when the crystal texture is parallel to the
SPO, i.e., in the y direction, the regions with relatively high
maximum principal stress increase noticeably for increas-
ing LPO and SPO (see Fig. 7b). This observation is
Fig. 6 Maximum principal
stress and its microstructural
standard deviation (the square
root of its variance throughout
the microstructure) as functions
of SPO and LPO for a
temperature change of ?100C.
The different LPOs, or crystal
texture ODFs, are denoted as
M1, M20 and M40, respectively
corresponding to a March–
Dollase parameter M of 1, 20,
and 40 for the distribution from
which the orientations were
chosen. The errors bars show
the standard deviation from five
independent replications of the
ODF. Three directions for the
LPOs were observed: x, y and z,
corresponding to the c-axes
having fiber texture along x-, y-,
and z-axis, respectively
Environ Earth Sci (2011) 63:1587–1601 1595
123
apparently counter to the results in Fig. 6, where the
average value of maximum principal stress for the elon-
gated fabric initially remains constant and then decreases
as M increases from 1 to 20 to 40. The explanation is in the
microstructural standard deviation of the maximum prin-
cipal stress, which increases as M increases from 1 to 20,
and only slightly decreases as M increases further from 20
to 40. This observation is related to the same phenomenon
described above for the strain energy density. By random
chance several of the elongated grains are crystallograph-
ically misoriented with respect to the general LPO.
Namely, these grains have large values of the fiber texture
angle h, which is not in concord with the expected prob-
ability distribution. Accordingly, the long direction of these
grains has a low coefficient of thermal expansion (a value
somewhat greater than that of the basal plane, i.e.,
-6.0 9 10-6C-1), while the textured matrix, in which
they lay, has an average coefficient of thermal expansion in
that direction that is somewhat less than that of the c-axis,
i.e., 26.0 9 10-6C-1. Thus, on heating the long direction
of these misoriented grain contracts, or only slightly
expands, in a direction where the remainder of the micro-
structure is greatly expanding. Hence, these misoriented
elongated grains have large axial tension. Therefore, even
though the average value of maximum principal stress for
the elongated fabric initially remains constant with crystal
texture and then decreases, these microstructures could
have a greater propensity for microcracking, since fracture
is determined by the extremes in the distribution, not by the
mean.
The stress state of these grains is better seen in Fig. 8,
where the stress components in the x and y directions, rxx
Fig. 7 Microstructural maps showing the spatial dependence of the
maximum principal stress for the three shape fabrics and the three
degree of crystallographic texture: a the crystal texture is perpendic-
ular to the shape fabric direction and in the plane of the microstruc-
ture(c-axes are textured along the x-direction); b the crystal texture is
parallel to the shape fabric direction (the c-axes are textured along the
y-direction); and c the crystal texture is perpendicular to the shape
fabric direction and out of the plane of the microstructure (the c-axes
are textured along the z-direction). Stresses are shown with a thermal
scale that ranges from black (0 MPa) to red-orange (125 MPa) to
white (C250 MPa)
1596 Environ Earth Sci (2011) 63:1587–1601
123
and ryy, respectively, are shown along with the maximum
principal stress. The thermal scale for the stress compo-
nents rxx and ryy ranges from black (B-250 MPa) to red-
orange (0 MPa) to white (C?250 MPa). The thermal scale
for the maximum principal stress is the same as that for
Fig. 7. From Fig. 8a, when the crystalline c-axis is textured
perpendicular to the SPO direction, grains with a relatively
high value of the maximum principal stress, the yellow-
Fig. 8 Microstructural maps of
the stress tensor components rxx
and ryy and the maximum
principal stress for the elongated
shape fabric, which has large
(M = 40) in-plane crystal
texture, a perpendicular to the
shape fabric (along the
x directions) and b parallel to
the shape fabric (along the
y direction). The stress tensor
components are shown with a
thermal scale that ranges from
black (B-250 MPa) to red-
orange (0 MPa) to white
(C?250 MPa). The thermal
scale for the maximum principal
stress is the same as that for
Fig. 7
Environ Earth Sci (2011) 63:1587–1601 1597
123
colored grains, are aligned along the crystal texture direc-
tion better than the average value. Thus, upon heating they
contract more (or expand less) than the average. Hence,
these well-aligned elongated grains have moderate axial
tension. On the other hand, highly misaligned grains, the
red- and black-colored grains, have large axial compres-
sion. For plane-stress simulations, the maximum principal
stress for a region with in-plane biaxial compression is the
zero out-of-plane stress, rzz. Thus, the highly misaligned
grains are not apparent in the maximum principal stress
microstructural maps.
From Fig. 8b, when the crystalline c-axis is texture
parallel to the SPO direction, the effect of misaligned
grains, as described above, is more striking. Clearly, the
grains with large values of the maximum principal stress
also have large values of ryy. The fact these highly stressed
grains were actually misoriented was validated for selected
grains by noting the misorientation angle h of these grains
and by computing the direction of the maximum principal
stress. So, while the average value of the maximum prin-
cipal stress decreases with LPO, the few misoriented grains
with large axial tension results in a large variance for the
maximum principal stress distribution and hence, the large
value of the microstructural standard deviation seen in
Fig. 6 for M20(y) and M40(y).
The different character of the microstructural maps for
the maximum principal stress when the crystal texture is
out of the plane of the microstructure can be seen in
Fig. 7c. These patterns result from the near isotropic in-
plane properties that result from this type of crystal
texture. Nonetheless, the patterns in the stress maps still
reflect the shape fabric. Additionally, the magnitude of
the maximum principal stress is reduced from that in
Fig. 7a, b.
These observations clearly show that both grain shape
fabric and crystal texture have significant impact on the
maximum principal stresses that arise in calcite micro-
structure due to temperature changes, and hence, on the
propensity for microcracking.
Coefficient of thermal expansion
For a given simulation configuration and a given temper-
ature change, the effective coefficient of thermal expansion
coefficient in the x direction, ax
eff, is computed from the
relative displacement change of the right and left sides of
the microstructure. Similarly, the effective coefficients of
thermal expansion coefficient in the y direction, ay
eff, is
computed from the relative displacement change of the top
and bottom of the microstructure. The results of these
measurements for the various combinations of SPO,
degrees of LPO, and the directional relationships between
them are given in Table 3. The uncertainty estimates rep-
resent the standard deviation from the five independent
replications of the ODF.
A distribution-averaged coefficient of thermal expansion
tensor in the sample reference frame can also be computed
from the crystal coefficient of thermal expansion tensor and
the March–Dollase probability distribution function. First
the crystal coefficient of thermal expansion tensor is rota-
ted to the sample reference frame using the orientational
relationship described by the polar angle, h, and the two
Azimuthal angles, u and x. Then, the rotated tensor is
averaged over all polar angles and all Azimuthal angles
weighted by the March–Dollase probability distribution
function, f(M, h) sin(h), where f(M, h) is given by Eq. (2).
The tensor values, thus computed, both parallel and per-
pendicular to the texture direction are given by:
Table 3 The effective coefficients of thermal expansion in the x- and
y-directions, ax
eff and ay
eff, in units of 10-6 C-1 for the three shape
fabrics (equiaxed, mixed, and elongated), the three degrees of LPO
(M1, M20, and M40), and the three directions of crystallographic
texture with respect to SPO (denoted by c–x, c–y, and c–z for when
the c-axes are aligned along x-, y- and z-direction, respectively)
Equiaxed Mixed Elongated
c–x c–y c–z c–x c–y c–z c–x c–y c–z
M1
ax
eff 4.1 ± 0.6 4.1 ± 0.6 4.1 ± 0.6 4.0 ± 0.5 4.0 ± 0.5 4.0 ± 0.5 4.1 ± 0.8 4.1 ± 0.8 4.1 ± 0.8
ay
eff 4.1 ± 0.6 4.1 ± 0.6 4.1 ± 0.6 4.0 ± 0.5 4.0 ± 0.5 4.0 ± 0.5 4.1 ± 0.8 4.1 ± 0.8 4.1 ± 0.8
M20
ax
eff 17.0 ± 0.5 -2.2 ± 0.1 -1.9 ± 0.4 17.0 ± 0.8 -1.9 ± 0.3 -1.6 ± 0.5 17.5 ± 0.8 -1.8 ± 0.6 -1.8 ± 0.6
ay
eff -2.0 ± 0.4 17.0 ± 0.5 -2.2 ± 0.2 -1.9 ± 0.5 16.5 ± 0.8 -2.2 ± 0.6 -2.3 ± 0.5 16.4 ± 1.0 -2.2 ± 0.5
M40
ax
eff 19.2 ± 1.0 -3.1 ± 0.5 -3.0 ± 0.4 19.6 ± 0.5 -3.1 ± 0.4 -3.0 ± 0.2 20.0 ± 1.1 -3.2 ± 0.5 -2.9 ± 0.7
ay
eff -3.1 ± 0.4 19.2 ± 1.0 -3.0 ± 0.5 -3.2 ± 0.2 19.3 ± 0.6 -3.2 ± 0.4 -3.3 ± 0.6 19.2 ± 1.3 -3.6 ± 0.4
1598 Environ Earth Sci (2011) 63:1587–1601
123
M ¼ 1 aeffparallel ¼ aeffperpendicular ¼ 4:67  106C1
M ¼ 20 aeffparallel ¼ 17:29  106C1
aeffperpendicular ¼ 1:64  106C1
M ¼ 40 aeffparallel ¼ 19:40  106C1
aeffperpendicular ¼ 2:70  106C1
As more crystal texture develops going from a random
ODF (M = 1) to a highly textured ODF (M = 40), the
effective coefficient of thermal expansion tensor
approaches the single-crystal tensor. The change of sign
in aperpendicular
eff from an expansion (a positive value) to a
contraction (a negative value) occurs at approximately
M = 8.30. Note, however, that the volume coefficient of
thermal expansion, given by the trace of the thermal
expansion tensor, is independent of crystal texture (i.e.,
M) and is equal to 14.0 9 10-6C-1.
The effective coefficients of thermal expansion in
Table 3 show a strong dependence on the degree of LPO
(i.e., M) and on the directional relationship between the
thermal expansion tensor and the crystal texture direction
(i.e., parallel or perpendicular), but only show a minor
influence on the shape fabric. The dependence of the
simulation results on the degree of LPO agrees well with
that computed for the distribution-averaged coefficient of
thermal expansion. Several minor deviations are apparent,
however, which are not fully understood at present. The
lack of dependence on shape fabric was unexpected. One
might have expected the shape fabric to have had a sig-
nificant influence on the effective thermal expansion when
combined with crystalline texture. For example, elongated
grains aligned in the crystalline texture direction might
have had more of an effect that an ensemble of equiaxed
grains. Simulation results seem to show that this is not the
case. However, as mentioned above, these effective values
represent only the contribution to the thermal expansion or
contraction from the elastic thermal misfit strains. If mi-
crocracking occurs, an additional expansion will occur,
which may depend on shape fabric and any synergy
between shape fabric and crystal texture.
Summary and conclusions
The elastic strain energy density and maximum principal
stress are important microstructural properties for predict-
ing microcrack formation. Spatial regions, where these two
microstructural properties are large are expected to indicate
regions in the microstructure with a propensity for micro-
cracking. Microstructure-based finite-element simulations
were used to elucidate not only these spatial regions, but
also the average values of these thermoelastic responses
and their microstructural standard deviations, i.e., the
standard deviation of their spatial distributions. This finite
element approach is considered an excellent tool for elu-
cidating influences of the rock’s fabric and crystal texture
on the thermoelastic behavior of marbles.
Significant observations are
• For random crystallographic texture, i.e., no LPO,
shape fabric did not have a significant influence on
either the average values or the microstructural stan-
dard deviation of the thermoelastic responses. While
not necessarily surprising, this observation was not
expected. There is, however, spatial dependence to the
stress networks that form. These networks of high strain
energy density and maximum principal stress mimic the
SPO, so this spatial dependence may have an influence
on microcrack formation.
• With equiaxed shape fabric, increasing degrees of LPO
generally lead to a reduction in the microstructural
stresses and the stored elastic strain energy. This effect
was expected since crystallographic texture reduces the
misfit strains from the thermal expansion anisotropy.
As expected, there is minor influence of the direction of
the crystal texture, since statistically there is no
directionality to the grain fabric.
• LPO in combination with a mixed or elongated shape
fabric can synergistically lead to significant effects,
which have a strong directional dependence.
• When the LPO is aligned parallel with the SPO
direction, the thermoelastic responses increase mark-
edly with shape fabric. The responses with crystal
texture are more complex, but essentially either
remain constant, increase slightly and then decrease,
or decrease. More significantly, however, is the
increase of the microstructural standard deviations of
these responses, indicating wide microstructural var-
iation of these responses. These microstructural vari-
ations are clearly apparent in the microstructural
response maps.
• When the LPO is aligned perpendicular to the SPO
direction, the thermoelastic responses remain constant
or decrease with shape fabric and decrease with crystal
texture. The microstructural standard deviations of the
strain energy density increase with shape fabric and
decrease with crystal texture. The microstructural
standard deviations of the maximum principal stress
decreases with both shape fabric and crystal texture.
Generally, microstructures with this combination of
shape fabric and crystal texture will have less of a
tendency to microcrack, particularly as the crystal
texture increases.
• When the LPO is aligned perpendicular to the SPO
direction and out of the plane of the simulation, the in-
Environ Earth Sci (2011) 63:1587–1601 1599
123
plane thermoelastic responses are nearly isotropic, and
are reduced in magnitude accordingly.
• Even though microstructures with aligned LPO and
SPO features give the largest thermoelastic responses
and microstructural variability, this response may not
be significant for several reasons. First, if the crystal
texture is correlated with the shape texture, these
misoriented grains may not be probable. Second, this
LPO is less often reported for calcite marbles. And
third, this combination of LPO and SPO would lead to
transgranular microcracking perpendicular to the elon-
gated direction of the grains, i.e., axial segmentation of
these grains. Such microcracking would therefore
depend on a low fracture energy cleavage plane, which
may not be the case.
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 helpful discussions. Financial support for E.R. Fuller at Go¨ttingen
University was provided by the Deutsche Forschungsgemeinschaft
with the 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 Noncommercial License which per-
mits any noncommercial use, distribution, and reproduction in any
medium, provided the original author(s) and source are credited.
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