Nonhydrostatic Stress State in the Martian Interior for Different Rheological Models

Abstract

We analyze nonhydrostatic stress field in the Martian interior for two types of heterogeneous elastic models: the ones with a lithosphere and the ones with a lithosphere and probable melting zones within it. Numerical modeling of the system of elastic equilibrium equations for a gravitating planet is carried out on a 1 × 1 arc degree spherical grid down to a depth of 1000 km. The boundary conditions are specified by the topography and gravity field data determined relative to a hydrostatic equilibrium spheroid taken as the reference surface. High maximum shear stresses in the zones of high tensile stresses are assumed as the criterion for selecting the probable marsquake sources. Irrespective of the type of rheological model, the zones of the high shear and tensile stresses in the crust and mantle of the Mars arc revealed beneath Hellas Planitia, Argyre Planitia, Mare Acidalia, Arcadia Planitia plain, and Valles Marineris.

STRESS CALCULATION ALGORITHM

The anomalous field of volume density $\delta \rho (r,\theta ,\varphi )$ is expanded in a series of spherical functions

$$\begin{gathered} \delta \rho (r,\theta ,\phi ) = \sum\limits_{i,n,m} {{{R}_{{inm}}}(r){{Y}_{{inm}}}(\theta ,\varphi )} \\ = \sum\limits_{i = 1}^2 {\sum\limits_{n = 2}^\infty {\sum\limits_{m = 0}^n {{{R}_{{i,n,m}}}(r){{Y}_{{i,n,m}}}(\theta ,\varphi )} } } . \\ \end{gathered} $$

((A1b))

For the convenience of solving the problem, the field is represented by the sum of infinitely thin layers located at different depths:

$$\delta \rho (r,\theta ,\varphi ) = \sum\limits_{j = 1}^N {\sum\limits_{i = 1}^2 {\sum\limits_{n = 2}^\infty {\sum\limits_{m = 0}^n {a_{{in,m}}^{j}} } } } {{Y}_{{i,n,m}}}(\theta ,\varphi ),$$

((A1b))

where amplitudes $a_{{i,n,m}}^{j} = {{R}_{{inm}}}({{r}_{j}})dr$ have the dimension [ML^–2], index j is the depth of the location of the anomalous gravitating layer, N is the number of layers,

$${{Y}_{{inm}}}(\theta ,\varphi ) = {{P}_{{nm}}}(\cos \theta )\left\{ \begin{gathered} \cos (m\varphi ),\,\,\,\,i = 1 \hfill \\ \sin (m\varphi ),\,\,\,\,i = 2 \hfill \\ \end{gathered} \right.,$$

and

${{P}_{{nm}}}(x)$ are the normalized Legendre functions

$$\begin{gathered} {{P}_{{nm}}}(x) = {{\left( {\frac{{2(n - m)!(2n + 1)}}{{(n + m)!}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}P_{n}^{m}(x),\,\,\,\,m \ne 0; \\ {{P}_{{nm}}}(x) = {{\left( {\frac{{(n - m)!(2n + 1)}}{{(n + m)!}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}P_{n}^{m}(x),\,\,\,\,m = 0, \\ {{P}_{{nm}}}(x) = {{(1 - {{x}^{2}})}^{{{m \mathord{\left/ {\vphantom {m 2}} \right. \kern-0em} 2}}}}\frac{d}{{d{{x}^{m}}}}({{P}_{n}}(x)). \\ \end{gathered} $$

((A2))

This spherical layer creates on the surface of a planet the gravity field

$$\Delta V = 4\pi GR\sum\limits_{i,n,m} {{{{\left( {\frac{r}{R}} \right)}}^{{n + 2}}}\frac{{{{R}_{{i,n,m}}}(r)}}{{(2n + 1)}}{{Y}_{{i,n,m}}}(\theta ,\varphi )} .$$

((A3))

Since the anomalous layer acts as a load on the planet, its interior experiences deformations leading to an additional perturbation of the potential. For obtaining the formula for the total anomalous potential that includes the global deformation of the planet, we introduce the term K_n(r) = (1 + k_n(r)) in (A3):

$$\begin{gathered} \Delta V = 4\pi GR \\ \times \,\,\sum\limits_{i,n,m} {{{{\left( {\frac{r}{R}} \right)}}^{{n + 2}}}\frac{{{{R}_{{i,n,m}}}(r)(1 + {{k}_{n}}(r))}}{{(2n + 1)}}{{Y}_{{i,n,m}}}(\theta ,\varphi )} , \\ \end{gathered} $$

((A4))

where k_n(r) (the load numbers of order n for a buried anomaly at a depth r) and K_n(r) (the Green function) were introduced in (Marchenkov et al., 1984; Zharkov et al., 1986).

In a similar way, we introduce numbers h_n—the load numbers describing deformation D(φ, λ) of the planet under load $\delta \rho (r,\varphi ,\lambda )\,:$

$$D(\theta ,\phi ) = \frac{{4\pi GR}}{g}\sum\limits_{i,n,m} {{{{\left( {\frac{r}{R}} \right)}}^{{n + 2}}}\frac{{{{R}_{{i,n,m}}}(r){{h}_{n}}(r)}}{{(2n + 1)}}{{Y}_{{i,n,m}}}(\theta ,\varphi )} .$$

((A5))

We represent the components of the displacement vector $\vec {u} = (u,v,w)$ in the spherical coordinate system

$$\begin{gathered} u = \sum\limits_{i,n,m} {{{U}_{n}}(r){{Y}_{{inm}}}(\theta ,\phi )} ,\,\,\,\,v = \sum\limits_{i,n,m} {{{V}_{n}}(r)\frac{{\partial {{Y}_{{inm}}}(\theta ,\phi )}}{{\partial \theta }}} , \\ w = \sum\limits_{i,n,m} {\frac{{{{V}_{n}}(r)}}{{\sin \theta }}\frac{{\partial {{Y}_{{inm}}}(\theta ,\phi )}}{{\partial \phi }}} . \\ \end{gathered} $$

((A6))

In the spherical coordinate system, variables y_i (i = 1, …, 6) are introduced:

$$\begin{gathered} {{y}_{1}} = U,\,\,\,\,{{y}_{2}} = \lambda X + 2\mu \dot {U},\,\,\,\,{{y}_{3}} = V, \\ {{y}_{4}} = \mu \left( {\dot {V} - \frac{V}{r} + \frac{U}{r}} \right),\,\,\,\,{{y}_{5}} = P, \\ {{y}_{6}} = \dot {P} - 4\pi G{{\rho }_{0}}U, \\ \end{gathered} $$

((A7))

where y₁, y₂, y₃, y₄, y₅, and y₆ are the radial terms in the normal displacement, in the normal stress (σ_rr), in the shear displacement, in the shear stresses (σ_rθ, σ_rφ), in the perturbation of gravitational potential, and in the gradient of perturbation of the potential minus the contribution due to radial displacement, respectively.

Problem (1)–(3) is reduced to the solution of a system of six first-order ordinary differential equations (see (Marchenkov et al., 1984)).

$$\begin{gathered} {{{\dot {y}}}_{1}} = - \frac{{2\lambda {{y}_{1}}}}{{(\lambda + 2\mu )r}} + \frac{{{{y}_{2}}}}{{\lambda + 2\mu }} + \frac{{\lambda n(n + 1){{y}_{3}}}}{{(\lambda + 2\mu )r}}, \\ {{{\dot {y}}}_{2}} = \left[ { - 4{{\rho }_{0}}{{g}_{0}}r + \frac{{4\mu (3\lambda + 2\mu )}}{{(\lambda + 2\mu )}}} \right]\frac{{{{y}_{1}}}}{{{{r}_{2}}}} - \frac{{4\mu {{y}_{2}}}}{{(\lambda + 2\mu )r}} \\ + \,\,\left[ {n(n + 1){{\rho }_{0}}{{g}_{0}}r - \frac{{2\mu (3\lambda + 2\mu )n(n + 1)}}{{\lambda + 2\mu }}} \right]\frac{{{{y}_{3}}}}{{{{r}^{2}}}} \\ \end{gathered} $$

$$\begin{gathered} + \,\,\frac{{n(n + 1){{y}_{4}}}}{r} - {{\rho }_{0}}{{y}_{6}}, \\ {{{\dot {y}}}_{3}} = - \frac{{{{y}_{1}}}}{r} + \frac{{{{y}_{3}}}}{r} + \frac{{{{y}_{4}}}}{\mu }, \\ {{{\dot {y}}}_{4}} = \left[ {{{\rho }_{0}}{{g}_{0}}r - \frac{{2\mu (3\lambda + 2\mu )}}{{(\lambda + 2\mu )}}} \right]\frac{{{{y}_{1}}}}{r} - \frac{{\lambda {{y}_{2}}}}{{(\lambda + 2\mu )r}} \\ \end{gathered} $$

((A8))

$$\begin{gathered} + \,\,\left\{ { - {{\rho }_{0}}{{\sigma }^{2}}{{r}^{2}} + \frac{{2\mu }}{{\lambda + 2\mu }}[\lambda (2{{n}^{2}} + 2n - 1) + 2\mu ({{n}^{2}} + n - 1)]} \right\} \\ \times \,\,\frac{{{{y}_{3}}}}{{{{r}^{2}}}} - \frac{{3{{y}_{4}}}}{r} - \frac{{{{\rho }_{0}}{{y}_{5}}}}{r},\,\,\,\,{{{\dot {y}}}_{5}} = 4\pi G{{\rho }_{0}}{{y}_{1}} + {{y}_{6}}, \\ {{{\dot {y}}}_{6}} = - \frac{{4\pi G{{\rho }_{0}}n(n + 1){{y}_{3}}}}{r} + \frac{{n(n + 1)}}{{{{r}^{2}}}}{{y}_{5}} - \frac{{2{{y}_{6}}}}{r}, \\ \end{gathered} $$

where $\lambda = K - \frac{2}{3}\mu .$

Using the Runge-Kutta method, we find the solution for each harmonics n up to the given degree. As was noted above, in this work we use the expansions of the topography and gravity field up to degree and order 90.

In the calculations of the conventional (surface) load numbers, anomalous densities are located on the surface of the planet, ${{a}_{{nm}}}(r) = {{a}_{{nm}}}(r = R).$ Let us write out the conditions of the problem for this case.

On the surface of the sphere r = R:

(a) the absence of shear stresses:

$${{y}_{4}}(R) = \mu \left( {\dot {V} - \frac{V}{r} + \frac{U}{r}} \right) = 0;$$

(b) the condition on the normal force due to the load:

$${{y}_{2}}(R) = \lambda X + 2\mu \dot {U} = - {{g}_{0}}(R){{a}_{{nm}}}(R);$$

(c) the boundary condition for the potential:

$${{y}_{6}}(R) + \frac{{(n + 1)}}{R}{{y}_{5}}(R) = 4\pi G{{a}_{{nm}}}.$$

Equations (A3) are integrated from the boundary of the outer liquid core, $r = {{r}_{c}},$ which is taken into account by the corresponding conditions on this boundary. These conditions have the following form:

(a) the absence of shear stresses

$${{y}_{4}}({{r}_{c}}) = 0;$$

(b) the condition of the normal force due to the load:

$${{y}_{2}}({{r}_{c}}) = {{\rho }_{{0i}}}({{r}_{c}})({{y}_{1}}({{r}_{c}}){{g}_{0}}({{r}_{c}}) - {{y}_{5}}({{r}_{c}});$$

(c) the boundary condition for the potential

$${{y}_{5}}({{r}_{c}})\gamma ({{r}_{c}}) + \frac{{(n)}}{{{{r}_{c}}}}\gamma - {{y}_{6}}({{r}_{c}}) = 4\pi G{{\rho }_{i}}{{y}_{1}}({{r}_{c}}).$$

Molodenskii’s function γ is determined by the structure of the core. In the case of a uniform core γ = 0.

In the case of a buried anomalous density $a_{{inm}}^{j}(r < R),$ the boundary conditions change.

On the surface, r = R:

$$\begin{gathered} {{y}_{4}}(R) = 0,\,\,\,\,{{y}_{2}}(R) = 0, \\ {{y}_{6}}(R) + \frac{{(n + 1)}}{R}{{y}_{5}}(R) = 0. \\ \end{gathered} $$

On the core–shell boundary, the conditions are the same as in the case of the surface loads provided that the anomalous densities are not located on this surface.

At the transition through the anomalous layer, at depths r_j, functions y₂ and y₆ have the following jumps:

(a) the discontinuity in the anomalous force:

$${{y}_{2}}({{r}_{j}} - 0) - {{y}_{2}}({{r}_{j}} + 0) = - {{g}_{0}}({{r}_{j}}){{a}_{{nm}}}({{r}_{j}});$$

(b) the break in the normal derivative of the potential:

$${{y}_{6}}({{r}_{j}} - 0) - {{y}_{6}}({{r}_{j}} + 0) = 4\pi G{{a}_{{nm}}}({{r}_{j}}).$$

The other functions y_i remain continuous. Thus, the elastic problem is completely defined.

The load coefficients (load Love numbers) are determined by the following formulas:

$$\bar {h}_{n}^{j} = \frac{{{{y}_{1}}(R)}}{R},\,\,\,\,\bar {k}_{n}^{j} = \frac{{{{y}_{5}}(R)}}{{R\bar {g}}} - 1,$$

where $R$ and $\bar {g}$ are the mean radius and mean gravitational acceleration on the surface.

Also, the load numbers are used:

$$h_{n}^{j} = \bar {h}_{n}^{j}{{\left( {\frac{{{{r}_{j}}}}{R}} \right)}^{{n + 2}}},\,\,\,\,k_{n}^{j} = (1 + \bar {k}_{n}^{j}){{\left( {\frac{{{{r}_{j}}}}{R}} \right)}^{{n + 2}}} - 1.$$

The solution of the system of equations (9) allows us to find the displacement field for each harmonic degree n and each depth j and then the field of the nonhydrostatic stress tensor.

The strain tensor components in the spherical coordinates have the following form:

$$\begin{gathered} {{\varepsilon }_{{rr}}} = \frac{{\partial u}}{{\partial r}},\,\,\,\,{{\varepsilon }_{{\theta \theta }}} = \frac{1}{r}\frac{{\partial v}}{{\partial \theta }} + \frac{u}{r}, \\ {{\varepsilon }_{{\phi \phi }}} = \frac{1}{{r\sin \theta }}\frac{{\partial w}}{{\partial \phi }} + \frac{v}{r}\cot \theta + \frac{u}{r}, \\ {{\varepsilon }_{{r\theta }}} = \frac{1}{2}\left( {\frac{{\partial v}}{{\partial r}} - \frac{v}{r} + \frac{1}{r}\frac{{\partial u}}{{\partial \theta }}} \right), \\ {{\varepsilon }_{{r\phi }}} = \frac{1}{2}\left( {\frac{1}{{r\sin \theta }}\frac{{\partial u}}{{\partial \phi }} + \frac{{\partial w}}{{\partial r}} - \frac{w}{r}} \right), \\ {{\varepsilon }_{{\theta \phi }}} = \frac{1}{2}\left( {\frac{1}{r}\frac{{\partial w}}{{\partial \theta }} - \frac{w}{r}\cot \theta + \frac{1}{{r\sin \theta }}\frac{{\partial v}}{{\partial \phi }}} \right). \\ \end{gathered} $$

((A9))

The stress tensor components are determined through the strain tensor components:

$$\begin{gathered} {{\sigma }_{{rr}}} = \lambda \Delta + 2\mu {{\varepsilon }_{{rr}}},\,\,\,\,{{\sigma }_{{\theta \theta }}} = \lambda \Delta + 2\mu {{\varepsilon }_{{\theta \theta }}}, \\ {{\sigma }_{{\phi \phi }}} = \lambda \Delta + 2\mu {{\varepsilon }_{{\phi \phi }}},\,\,\,\,{{\sigma }_{{r\theta }}} = 2\mu {{\varepsilon }_{{r\theta }}}, \\ {{\sigma }_{{r\phi }}} = 2\mu {{\varepsilon }_{{r\varphi }}},\,\,\,\,{{\sigma }_{{\theta \phi }}} = 2\mu {{\varepsilon }_{{\theta \phi }}}, \\ \end{gathered} $$

((A10))

where Δ is dilatation:

$$\begin{gathered} \Delta = \sum\limits_{i = 1}^2 {\sum\limits_n {\sum\limits_m {{{X}_{n}}(r){{Y}_{{inm}}}(\theta ,\phi )} } } , \\ {{X}_{n}}(r) = {{{\dot {U}}}_{n}}(r) + \frac{2}{r}{{U}_{n}}(r) - \frac{{n(n + 1)}}{r}{{V}_{n}}(r), \\ \end{gathered} $$

((A11))

where the dot denotes differentiation with respect to r, θ is the polar angle, ϕ is longitude, and λ = K–2/3μ is the Lame parameter.

Substituting (A6) and (A9) into (A10) and summing up the harmonic series, we obtain the stress tensor components ${{\sigma }_{{ik}}}$ to the given degree n. The arbitrary additional nonhydrostatic stress state at the point under consideration is characterized by the extension or compression of the vicinity of the point in three mutually perpendicular directions. At each point (r, θ, ϕ), the symmetric total stress tensor σ_ik is reduced through the transformation of the coordinates to the diagonal form, which makes it possible to obtain the respective normal stresses ${{\sigma }_{1}},$${{\sigma }_{2}},$ and ${{\sigma }_{3}}.$ The principal stresses σ_{3 ≤} σ_{2 ≤} σ₁ are found as the roots of the cubic equation $\left| {{{\sigma }_{{ik}}} - {{\sigma }_{k}}{{\delta }_{{ik}}}} \right| = 0,$i = 1, 2, 3.

The flow chart of the calculations is shown in Fig. A1.

Fig. A1.

Flow chart of stress calculation in Mars interior.