Raman elastic thermobarometry has recently been applied in many
petrological studies to recover the pressure and temperature (

The effect of changing the aspect ratio on residual stress is investigated,
including quartz, zircon, rutile, apatite, and diamond inclusions in garnet
host. Quartz is demonstrated to be the least affected, while rutile is the
most affected. For prolate quartz inclusion (

Raman elastic thermobarometry has been extensively used to recover the
pressure and temperature (

For an ellipsoidal, elastically anisotropic inclusion entrapped in an infinite isotropic host, an exact closed-form analytical solution has been available for a long time (Eshelby, 1957; Mura, 1987). This solution has been widely applied in Earth science for many problems, such as viscous creep around inclusions (Freeman, 1987; Jiang, 2016), flanking structures (Exner and Dabrowski, 2010); elastic stress of inclusions at various scales (Meng and Pollard, 2014), microcracking and faulting (Healy et al., 2006), and magma-chamber-induced deformations (Bonaccorso and Davis, 1999). The advantage of such an approach is that no numerical software or programming is required and the solution can be obtained rapidly and precisely, with no numerical approximation error. The rapid calculation also permits in-depth, systematic stress and strain analysis of the inclusion–host system or a potential Monte Carlo simulation for uncertainty propagation. The procedure of calculating the residual stress in an ellipsoidal anisotropic inclusion embedded in an elastic, isotropic host is based on the equivalent eigenstrain method and the classical Eshelby solution (Eshelby, 1957; Mura, 1987). Recently, the Eshelby solution has been applied to exhumed mineral inclusions entrapped in a host and the result has been compared to the finite-element method (Morganti et al., 2020). Mineral inclusions were measured for their crystallographic orientation and shape via X-ray diffraction and tomography (Morganti et al., 2020), but the significance of the aspect ratio, shape, and crystallographic orientation has not been studied in a systematic way. More importantly, the Eshelby solution only applies to perfectly ellipsoidal inclusions, but natural mineral inclusions are faceted. Therefore, the uncertainty in and limitations of using the Eshelby solution for natural faceted inclusions remain to be investigated. In this study, we explore the Eshelby solution in-depth relating to the inclusion–host problem. A general analytical form is first presented in this paper (the previous submission record is given in the Acknowledgements) following the Eshelby equivalent eigenstrain method (Mura, 1987, chap. 4) to calculate residual stress and strain of an anisotropic inclusion in an isotropic host. For inclusions belonging to certain crystallographic symmetry, such as tetragonal, hexagonal, and trigonal classes, simplified explicit expressions describing residual stress and strain are derived. The analytical formulas are cross-validated against the numerical results obtained using a self-developed finite-element code. Convergence tests are successfully performed to show the correspondence of the numerical (FE) and analytical solutions. In-depth analysis of the effects due to (1) inclusion elastic anisotropy, (2) inclusion aspect ratio, and (3) relative orientation between the inclusion crystallographic and geometrical principal axes are performed to show how they affect the application of elastic thermobarometry. The MATLAB code has also been made available alongside the submission.

One major problem of using the Eshelby solution for mineral inclusion is that natural inclusions are faceted in shape, which leads to a heterogeneous residual stress field (e.g. Chiu, 1978; Mazzucchelli et al., 2018). To resolve this issue, we use our self-developed 3D finite-element code to simulate the residual stress distribution within faceted inclusions of varying shapes. Fitting an arbitrary 3D shape with effective ellipsoid is a common practice in image analysis and microstructural research (e.g. Ghosh and Dimiduk, 2011). We explore the possibility of using an effective ellipsoid to approximate the shape of a faceted inclusion. The residual stress obtained from the analytical solution based on the best-fitted effective ellipsoid is used as a proxy to represent the volumetrically averaged stress within the faceted inclusion. By inspecting the numerical (FE) and analytical solutions, we have found that for most mineral inclusions, e.g. quartz, zircon, apatite, and rutile, the volumetrically averaged stress represents the stress state of arbitrarily faceted inclusions very well. This may potentially provide a useful guide for future applications of elastic thermobarometry for any natural faceted mineral inclusions.

We consider an anisotropic, ellipsoidal solid inclusion entrapped in an
isotropic, infinite host at high

To simulate the exhumation of the inclusion–host system to room

Schematic diagram showing how to obtain the residual stress and strain
of anisotropic inclusion in isotropic host.

Because mechanical equilibrium is not satisfied for the stressed inclusion
embedded in a stress-free host, elastic deformation will occur (stage shown
in Fig. 1b–c). The amount of elastic deformation that affects the
inclusion with a pre-strain

Eshelby's solution treats a homogeneous, ellipsoidal, isotropic
inclusion embedded in an infinite isotropic host (Eshelby,
1957). Following Eshelby (1957), we replace the inclusion–host system with an
isotropic homogeneous space with an elastic tensor

All the other components are zero. For general ellipsoidal inclusions, the

Following the equivalent eigenstrain method (Mura, 1987,
chap. 4), one may appropriately choose the equivalent eigenstrain

By doing so, the stresses in the original anisotropic heterogeneity and the
equivalent isotropic inclusion will be equal. This is because the host is
stressed (and strained) by the same amount following Eq. (6), which leads to
the same inclusion stress because the traction is matched between inclusion
and host. By replacing the strain

The equivalent eigenstrain

The dimensionless matrix

The wavenumber shifts of Raman peaks are induced by lattice strain. By
measuring wavenumber shift of the inclusion in a thin section, it is
possible to recover the residual strain preserved within the inclusion
(Angel et al., 2019; Murri et al., 2018).
This can be done by using the Grüneisen tensor. Therefore,

By inverting the right-hand matrix in Eq. (8), the eigenstrain terms can be
expressed as a function of residual strain

For tetragonal or hexagonal minerals, e.g. zircon, rutile, and apatite, the
stiffness tensor comprises six independent components:

By substituting the stiffness tensor components and the measured residual strains, the eigenstrains can be directly calculated. The equation above thus allows for the estimation of the entrapment (hydrostatic or non-hydrostatic) stress (or strain) conditions by knowing the residual stress and strain conditions of the inclusion.

We have validated our implementation of the proposed analytical framework
against independent finite-element (FE) solutions. A self-written 3D FE code
is used to validate the presented analytical solution
(Dabrowski et al.,
2008; Zhong et al., 2018). For validation purposes, we used spheroidal
quartz inclusions in an almandine garnet host. Adaptive tetrahedral
computational meshes, with the highest resolution within and around the
inclusion, are generated with Tetgen software (Si, 2015). The
anisotropic elastic properties of quartz inclusion at room

In Fig. 2a, the numerically and analytically obtained residual stresses are
plotted together as a function of the aspect ratio of the tested spheroidal
inclusions. In Fig. 2b, the difference is plotted as a function of element
count and boundary distance (

Cross-validation results between a finite-element method and the
presented analytical method.

In addition, we have also tested the effect of applying cubic elastic stiffness tensor of almandine from Jiang et al. (2004) and compared the residual stress with the results obtained for an elastically isotropic garnet (blue dots in Fig. 2a). The difference in residual stresses obtained with FE method using anisotropic garnet host and the analytical solution (implicitly assuming isotropic host) is less than 0.001 GPa. This suggests that it is not necessary in practice to consider the anisotropy of garnet host. This has been also reported in, e.g. Mazzucchelli et al. (2019). It was suggested that the elastic anisotropy of cubic garnet has no significant impact on the result of elastic barometry. Thus, effective isotropic elastic properties of garnets may be used to model the inclusion–host elastic interaction.

In Eq. (8), the aspect ratio of the inclusion only affects the Eshelby tensor.
Here, we choose some common inclusions in metamorphic garnets, including
quartz, apatite, zircon, and rutile, as examples to test the effect of
inclusion aspect ratio on residual stresses. The data sources of elastic
stiffness tensors of the studied minerals are listed in the caption of Fig. 3. Here, we first focus on spheroidal inclusions and let the
crystallographic

Effect of geometrical aspect ratio of spheroidal inclusion along

To isolate the effect of the inclusion aspect ratio, the eigenstrains for
various inclusion minerals are all set to create 1 GPa compressive
hydrostatic residual stress for the reference spherical inclusion embedded
in an isotropic, infinite garnet host, which is the same approach as in the
previous “cross-validation” section. Therefore, the stress variations
shown in Fig. 3 only represent the mechanical effect due purely to varying
the geometrical aspect ratio

Among all the tested minerals, the residual stress in quartz inclusions is
the least sensitive to variations in aspect ratio. For prolate quartz
inclusion (

The residual stress in rutile is the most sensitive to aspect ratio
variations. With increasing

The pressure (negative mean stress) is significantly less sensitive to
inclusion aspect ratio variations. For prolate inclusions of quartz, zircon,
and apatite, the residual pressure differs from the reference level
(spherical inclusion shape) by only ca. 0.01 GPa when the aspect ratio

The residual stress in mineral inclusions can be easily converted into
residual strain, which can be directly translated into Raman shifts
(Angel et al., 2019; Murri et al., 2018).
The effects of varying the aspect ratio of a quartz inclusion on Raman
wavenumber shifts (see Fig. 4) are determined using the calculated residual
strain components and the Grüneisen tensor
(Murri et al., 2018). The same model settings are
applied, with 1 GPa compressive hydrostatic residual stress characterizing
the case of a spherical quartz inclusion. It is shown that for prolate
inclusions, aspect ratio only introduces minor effects on the Raman shifts.
For example, varying the

The effect of geometrical aspect ratio of ellipsoidal quartz
inclusion for

Our results show good consistency with the Raman data reported in Kouketsu et al. (2014), where quartz inclusions with different aspect ratios were measured and no significant variation in spectral shift was found.

In nature, the crystallographic axes of an inclusion are not necessarily
aligned parallel to its geometrical axes. In this section, the effect of
varying the crystallographic orientation with respect to the geometrical
axes on the resulting Raman wavenumber shift is systematically studied using
the proposed analytical model. Here, we reorient the crystallographic

The effect of varying the crystallographic orientation (

The results are shown in Fig. 5. For an aspect ratio of 2, the 464 cm

The analytical solution presented in this study is derived for an ellipsoidal anisotropic inclusion. However, the shape of a natural mineral inclusion may exhibit corners, edges, and facets, which results in stress concentration effects and may have an impact on the overall level of the residual stress.

Here, we explore the possibility of using an effective ellipsoid to fit the shape of a faceted inclusion. We use the equivalent aspect ratio to calculate the residual stress or strain based on the presented analytical solution for ellipsoidal inclusions. Fitting an arbitrary, irregular shape using an ellipsoid in 3D (or an ellipse in 2D) is a common practice in image analysis (Chaudhuri and Samanta, 1991; Li et al., 1999). A pixelated 3D image is used to calculate the second-order moment of the object shape to minimize the mismatch between the 3D irregular inclusion shape and the effective ellipsoid. The method allows for obtaining the lengths and orientations of the major, minor, and intermediate axes of the effective ellipsoid (the method described in Appendix A–C and MATLAB code provided in the Supplement can be used to perform this task).

Similar to previous sections, we use the eigenstrain components that can
load the reference spherical inclusion of any given mineral embedded in
isotropic almandine garnet host into 1 GPa compressive residual stress. The
tested inclusion shapes include: cylinder, tetrahedron, cuboid, octahedron,
hexagonal prism, and icosahedron. To vary the aspect ratio, the inclusion
shape is stretched in the

We study five inclusion minerals: quartz (elastic tensor from Heyliger et al., 2003), zircon (Bass, 1995), rutile (Wachtman et al., 1962), fluorapatite (Sha et al., 1994), and diamond (Bass, 1995). Almandine garnet is taken as the host grain (Milani et al., 2015). For each FE mesh, the size of the computational box is set more than 10 times the inclusion size and adaptive mesh is generated with highest mesh resolution within and close to the inclusion. The 10-node tetrahedron elements with quadratic (second-order) shape (interpolation) functions for the displacement field are used. In total, there are ca. 2 million elements per model (relative error of less than 0.0003 based on benchmark results in Fig. 2).

The results of numerical simulations are shown in Fig. 6. The effective aspect ratio for all different inclusion shapes, together with the residual stress components, are given in the Supplement. The residual stress in non-ellipsoidal inclusions based on the FE model is heterogeneous and we monitor the stress state (1) at the centroid (CT) point (red dots in Fig. 6) and (2) as the volumetric average (VA) within the entire inclusion (orange dots)

The root-mean-square deviation (RMSD) is calculated by comparing the residual stress from the finite-element solutions based on various stress evaluation schemes (CT and VA) and analytical models using the best-fitted effective aspect ratio (Table 1). It is clearly shown that the VA stresses of quartz, zircon, rutile, and apatite inclusion are remarkably similar to the analytical results, with an RMSD generally lower than 0.02–0.03 GPa (ca. 2 %). From CT to VA, a significant improvement on RMSD of a factor of abour 2 is obtained.

Finite-element stress of various inclusion shapes (symbols) compared
to the stress of effective spheroidal inclusion based on an analytical method
(black curves). The effective aspect ratio of inclusion shape is calculated
based on the fitting method of
Chaudhuri and
Samanta (1991) and Li et al. (1999) (see Appendix A–C). The inclusion is loaded
with eigenstrain that generates 1 GPa compressive hydrostatic residual
stress for spherical shape. The variation of stress is only caused by the
shape change. The

Root-mean-square deviation (RMSD) of the finite-element solution of symmetrically shaped non-ellipsoidal inclusion in Fig. 6 compared to the analytical solution of equivalent spheroidal inclusions. Isotropic almandine garnet is used as a host. For each inclusion mineral and inclusion shape, the aspect ratio varies from 0.2 to 5. The effective aspect ratio is calculated for each shape and used for the analytical solution to obtain the residual stress state. The inclusion is loaded by eigenstrain that creates 1 GPa hydrostatic residual pressure for spherical inclusion in an infinite host. Thus, any stress variation can only be caused by shape change. The calculated stress data for each individual FE run is given in Supplement data. Stress is obtained for (1) the centroid point (CT) and (2) volumetric average (VA) of the entire inclusion (see Fig. 6 for illustration). The RMSD is calculated by comparing the FE results and analytical results based on the best-fitted effective ellipsoid. The unit used for RMSD is GPa. The elasticity of the inclusion mineral given in the caption of Fig. 6.

The only exception among the studied minerals is diamond, where the RMSD is
higher than in the case of other inclusions, which are elastically softer.
This is consistent with the high “geometrical correction factor” reported
for diamond in Mazzucchelli et al. (2018). However, as an
improvement from Mazzucchelli et al. (2018), where the
geometrical correction factor must be applied for all inclusion phases to
correct the residual stress due to shape effects, we have found that the
stress variation due to varying inclusion shapes for minerals such as quartz,
zircon, apatite, and rutile can be satisfactorily approximated by using the
proposed approach of the equivalent ellipsoidal inclusion, with RMSD
generally lower than 3 %–4 % for most of the studied inclusion shapes. To
achieve this improved and satisfactory level of prediction (1) we have used
best-fit ellipsoids to better approximate inclusion shapes, instead of a
crude measure of the length/width ratio of, e.g. cuboidal or cylindrical
inclusion, and (2) we have not only considered the centroid point of the
inclusion (which indeed yields a larger RMSD) but also the volumetric
average (VA) for the residual stress state sampled during stress
measurements, which interestingly provides a significantly better
approximation for the residual stress or strain state of the tested mineral
inclusions. This is practical and useful in Raman measurement because it is
possible to perform either (1) multiple-point averaging during Raman analysis
within the entire inclusion or (2) defocus the laser beam to take into
account a larger volume for the inclusion strain heterogeneity. For tiny
inclusions (size of ca. 1–3

In nature, the shape of mineral inclusions is not necessarily as highly
symmetric as presented in the previous section, and the crystallographic orientation can be generally random with respect to the principal geometric
axes of the inclusion. Here, a MATLAB script is used to generate completely
random 3D inclusion shapes by prescribing random vertices (non-coplanar 5 to
24 vertices) and connecting them to form a closed 3D shape. Delaunay
triangulation is used to form 3D volumes enclosed by the triangular faces.
Tetgen software is again used to generate unstructured tetrahedron
computational meshes fitted to the inclusion surface. The effective aspect
ratio (geometrical longest to shortest axis of the best-fitted effective
ellipsoid) is controlled to be within 6. In total, we have generated ca. 500
random 3D inclusion shapes and performed a finite-element simulation for the
previously studied set of anisotropic inclusion minerals (quartz, zircon,
rutile, apatite, and diamond) to calculate the elastic stress field. The
generated random shapes are plotted in the Supplement (see the .gif
animation that illustrates 100 selected examples of 3D inclusion shapes). We
further allow the crystallographic

The 500 randomly generated inclusion shapes (top panel for examples)
calculated with the finite-element method (vertical axis) and analytical method
(horizontal axis). All three normal stress components are plotted together
in each diagram. The crystallographic

For the centroid point (CT), quartz inclusions have the lowest RMSD (ca. 0.03 GPa) for all three normal stress components and diamond inclusions have the highest RMSD (ca. 0.11 GPa). In general, CT stress shows a systematically higher RMSD than volumetrically averaged (VA) stresses. When the stress within the inclusion is volumetrically averaged, the RMSD dramatically drops to a nearly perfect match between the FE results for irregularly faceted inclusion and the analytical prediction based on the best-fitted ellipsoid. The RMSD of volumetrically averaged residual stresses (VA) of quartz, zircon, rutile, and apatite are all lower than ca. 0.02 GPa (2 %), and it shows no obvious dependence on the effective aspect ratio even for the extremely elongated or flattened inclusions (see the near-perfect alignment of the orange dots and 1-to-1 ratio line in the middle and bottom panels of Fig. 7).

Thus, the volumetric average of the residual stress within the inclusion provides a sufficiently reliable match between the exact results for irregularly shaped inclusions and the approximate predictions based on the analytical solution. This shows that it is possible to approximate the stress and strain state of the inclusion using an effective ellipsoid shape for the tested inclusions including quartz, zircon, rutile, and apatite. Only diamond has a notably higher level of RMSD, exhibiting a systematic discrepancy between the exact numerical and approximate analytical results. This indicates that using the proposed equivalent analytical model for diamond inclusion may lead to a potential overestimation of the residual stress by ca. 0.07 GPa (7 %). However, this RMSD may still be acceptable as a crude estimate or may serve as an upper limit for elastic thermobarometry.

The presented model builds on a linear elastic constitutive law at room
temperature, i.e.

In this study, we use the classical Eshelby solution combined with the
equivalent eigenstrain method to calculate the residual strain and stress in
an anisotropic, ellipsoidal mineral inclusion embedded in an elastically
isotropic host. The residual stresses can be expressed by a linear operator
(Eq. 8) acting on the eigenstrain. The linear operator depends on the
anisotropic elastic stiffness tensor of the inclusion evaluated at room

The effect of inclusion aspect ratio on the inclusion residual stress and
strain has been investigated quantitatively. The residual stress in quartz
inclusions exhibits the least sensitivity to aspect ratio changes, and rutile
shows the most pronounced variation. The popularly used quartz-in-garnet
system is studied in more detail. For prolate quartz inclusions, the
residual stress variations caused by varying inclusion shape are shown to be
insignificant when the crystallographic

Our proposed analytical procedures to model residual inclusion stress and
strain state do not require pre-FE simulation to obtain the six-by-six
“relaxation tensor” as proposed by Mazzucchelli et al. (2019). For
application purposes, as long as the lattice strains of inclusion
(

The presented model is only exact for perfectly ellipsoidal inclusions. In
nature, inclusions often possess different shapes with facets and edges.
Finite-element simulations on various faceted inclusion shapes showed that
the residual stress is modified to a different degree compared to the
simple ellipsoidal inclusion case, depending on the relative elastic
properties between the inclusion and the host grain. However, the proposed
approach of using the analytical result for the best-fitted effective
ellipsoids yields remarkably good approximation for all the tested inclusion
shapes, including highly irregular shapes. The RMSD comparing the FE numerical
solution for faceted inclusion and the analytical solution based on a
effective best-fitted ellipsoid is typically less than 2 % for quartz,
zircon, apatite, and rutile inclusions. The only exception are the
elastically stiff diamond inclusions, where the RMSD reaches 7 %. This
finding expands the applicability of the analytical framework to arbitrarily
shaped inclusions, whose elastic stiffness is not significantly higher than
host (such as quartz, rutile, zircon and apatite). One important
petrological implication is that it is possible to take the volumetrically
averaged stress and strain within the inclusion and use it as a proxy to
represent the residual stress and strain state of the inclusion. Then the
proposed analytical framework may be used to recover the entrapment
condition by back-calculating the eigenstrain using the volumetrically
averaged residual stress and strain and the effective ellipsoid aspect ratio of
the inclusion (Eq. 8). In fact, averaging the stress and strain within a certain
volume is implicitly done in practical Raman measurement, e.g. for a
tiny micrometre-sized inclusion with a laser beam size typically
exceeding 1

When the inclusion and host are crystalized at entrapment conditions, they are
considered to be stressed and strained by taking the room

The matrix

A MATLAB code is provided to perform this calculation
(

The components of Eshelby's tensor

The integrals

The method (for details,
see Chaudhuri and Samanta, 1991; Li et al., 1999) requires a 3D data image
of the inclusion consisting of regular cubic voxels that has volume

All MATLAB code has been uploaded in the Supplement. Details for how to use the code can be found in Appendix A–C or by directly contacting the authors at xinzhong0708@gmail.com.

All calculated data are available as an excel file (provided in the Supplement) with details provided in Appendix A–C.

The supplement related to this article is available online at:

XZ and MD conceived the idea and developed the analytical method. XZ created the finite-element model. XZ, MD, and BJ wrote the manuscript together.

The authors declare that they have no conflict of interest.

This work was finished during 2017–2018 and the paper
was submitted to

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. P2EZP2_172220), the Pánstwowy Instytut Geologiczny – PIB (grant no. 62.9012.2063), and the European Research Council, H2020 European Research Council (grant no. DIME (669972)).We acknowledge support from the Open Access Publication Initiative of Freie Universität Berlin.

This paper was edited by Taras Gerya and reviewed by two anonymous referees.