Data-driven analytical mechanics of aging viscoelastic shotcrete tunnel shells

Based on the principle of virtual power, equilibrium conditions are established for the forces within a cross section of a tunnel top heading driven according the New Austrian Tunneling Method (NATM). External forces, namely impost actions and ground pressure distributions following a third-order polynomial, are analytically linked with internal forces, such as axial forces and bending moments, arising as integrals over the shell thickness, of circumferential normal stresses. The latter are related, via an aging viscoelastic shotcrete material model, to circumferential normal strains, as well as to radial and circumferential displacement components. This allows for analytical transformation of displacement measurement data collected at the crown and the footings of the shell segment, into ground pressure and impost action evolutions, together with all the associated force and stress quantities. For the Sieberg tunnel, driven in Miocene clay marl, our data-driven analytical mechanics model evidences virtually uniform ground pressure distributions, leading to a first rapidly increasing, and then mildly decreasing utilization degree of the shotcrete shell.


List of symbols
Level of loading associated with the strength of the material P int Virtual power of internal forces P ext Virtual power of external forces × Exterior product (cross-product) of two vectors in 3D Euclidean space · Inner product of two vectors in 3D Euclidean space : Second-order tensor contraction ⊗ Dyadic product of two vectors ∇ Nabla operator ∂ Symbol used to denote partial derivatives d Symbol used for differentials

Introduction
As a rule, engineering structures are designed for a priori known, standardized external loads they have to withstand. This design strategy is heavily challenged in tunnel engineering where the loads acting on the tunnel shells are not known a priori: Since these loads are tightly linked to the interplay of the tunnel shell and the surrounding soil or rock, they heavily depend on geological details which remain unknown until the very construction phase. During the latter, the aforementioned interplay is standardly monitored in terms of displacement measurements, traditionally in terms of convergences [5,23]; and more recently in terms of laseroptical geodesy [29,30]. Hence, as a rule, both the loads acting on the tunnel shell and the stress states in the latter, need to be estimated from the aforementioned displacement measurements. This is very often done in the context of the so-called convergence-confinement method [3,5,22,24,31,32]. It reduces the ground-shell interactions to a plane-strain axisymmetric problem, with the tunnel radius change ("convergence") and the radial pressure acting at the shell-ground interface ("ground pressure") as the key physical properties. While this problem is particularly helpful for closed circular tunnel shells, its use for the construction phase where only part of the tunnel shell is installed, and where, consequently, the "ring closure" is not realized yet, is more questionable; and extensions for non-symmetric cases are desirable. This is the very focus of the present contribution, dealing with (non-closed) top headings made of hydrating shotcrete in the framework of the New Austrian Tunneling Method (NATM), and monitored by means of geodetic displacement measurements of selected points on the inner surface of the tunnel shell. Rather than coming up with a mechanical model for the ground around the tunnel, we will explicitly incorporate the latter measurements into a "hybrid analysis" in the line of [2,11,12,19]. In this context, we will explicitly introduce the outer surface of the tunnel shell onto which normal traction forces ("ground pressure") will be imposed. Equilibrium of forces within each cross-sectional plane of the tunnel shell will be maintained by normal tractions acting on the footings of the tunnel shell segments.
The corresponding extension of the convergence-confinement methods is developed on the basis of key theoretical ingredients of continuum mechanics: (i) the principle of virtual power [6][7][8][9][10]35,40] applied to thin cylindrical shell segments (with the virtual motions being restricted to the cross-sectional plane of the tunnel shell), and (ii) multiscale aging viscoelasticity [28] quantifying the constitutive behavior of shotcrete.
Accordingly, the paper is organized as follows: We first recall the principle of virtual power, so as to develop an analytical circular arch theory governing the equilibrium of forces within the cross-sectional planes (Sect. 2). Thereafter, we transform power-law creep functions associated with uniaxial stress experiments, into shell-specific relaxation functions considering free inner shell surfaces and plane strain conditions (Sect. 3). Corresponding mathematical relationships are then combined in order to come up with elaborate analytical formulae linking displacement measurements at three geodetic laser-optical targets, with traction forces acting on a circular NATM tunnel heading (Sect. 4). These analytical tools are applied to the measurements performed at the Sieberg tunnel -a benchmark example having undergone various assessment strategies [12,37], before the results are discussed, so as to conclude the paper (Sect. 5).
2 Cross-sectional equilibrium within an arch-like tunnel cross section, derived from the principle of virtual power

Starting point-equilibrium of forces within tunnel cross section
According to the principle of virtual power [6-10,35] a mechanical system is in equilibrium (i.e., it fulfills the momentum and angular momentum conservation laws for the special case of negligibly small accelerations), if the power performed by the external and internal forces on an arbitrary virtual velocity field vanishes. We note that a mathematically identical principle is sometimes called principle of (rates of) virtual work; then the virtual velocities are called virtual motions [26,27].
We specify the principle of virtual power for the forces being equilibrated within a particular cross section of the tunnel, with a cross-sectional area A and a contour (boundary) C, see Fig. 1. All positions within such a cross section are labeled by vectors x, and any continuous and differentiable velocity fieldv(x) with vectors lying within the cross-sectional plane qualifies as admissible virtual velocity field. Then, the power density of external forces, per unit length measured along the tunnel axis, reads as with the volume forces f, and traction forces T, and n as the outward normal onto a surface element dC. The power density of internal forces is constructed on the simplest velocity-derived quantity which is objective, i.e., observer-independent. This is the Eulerian strain rate tensor Accordingly, we have [9] whereby σ is the symmetric second-order Cauchy stress tensor -any non-symmetric tensor portion would not perform any power ond. Mathematically, the principle of virtual power then reads as Application of the chain rule and of the divergence theorem to (4), together with (1) and (2), then readily yields [9] ∀x ∈ A : divσ + f = 0, (5) ∀x ∈ C : σ · n − T = 0 ( 6 ) whereby all vectors occurring in (5) and (6) are restricted to the cross-sectional plane.

Virtual velocity field characterizing a slender circular tunnel shell segment
The principle of virtual power is particularly suitable tool for deriving theories for structural components, such as beams, arches, or plates [13,16,34,40]. In this context, the virtual velocity field governing the equilibrium conditions is further restricted; and this yields differential equations of lower dimensions than those of (5) and (6). In the case of circular cylindrical shell segments, the following characteristics are considered: The circular arch is geometrically defined by a segment of a circle with radius R (the arch line) and by the thickness h, see with r and ϕ standing for the radial and circumferential polar coordinate, respectively; e r (ϕ), e ϕ (ϕ), and e z are the unit vectors spanning a corresponding orthonormal frame. More precisely, ϕ is the angle which is measured counterclockwisely from the base vector e x of a Cartesian base frame made up by the base frame e x , e y , and e z and the origin O, see Fig. 1. The radius R and the polar angle ϕ give access to the arc length s, Each generator line of the circular slender arch is characterized by the following rigid body motions, see Fig. 2: -it undergoes a translational motion orthogonal to the circle within the plane of the circle, quantified by a virtual velocity e rv C r , withv C r being the radial component of the virtual velocity of the gravitational center of the generator line; -it undergoes a rotational motion around the center of the circle (i.e., around a vector directed in e z and positioned in the origin "O" of the coordinate system seen in Fig. 1), quantified by a virtual velocity e ϕv where we also made use of (8). Conclusively, the virtual velocity field governing the equilibrium of the circular arch segment reads asv However, for the development of structural theories of stretching and bending members, it is advisable to introduce e z -related rotations around the gravitational center of the generator (associated with bending) and translations in tangential direction (associated with stretching). Accordingly, we consider the rotation around the center of the circle as a superposition of a tangential translation of the gravitational center of the generator and a rotation around the latter center; mathematically, this reads aŝ see the sketches at the right handside of Fig. 2.

Virtual power of internal forces-axial forces and bending moments in the tunnel shell
The components of the Eulerian strain rate tensor in a polar base frame read as, see e.g., [26, p. 747] and insertion of (10) and (11) into (12.2) yieldŝ Specification of the internal power expression (3) for the shell-specific virtual strain rates (13) and (14) yields whereby ϕ b and ϕ e are the values of the polar angle at the beginning and at the end of the tunnel segment, respectively. Taking all terms independent of r out of the respective integral, expression (15) for the virtual power of the internal forces can be transformed to whereby the separate integration over r has induced the new, shell-specific internal forces called axial forces (per length) and bending moments (per length) It becomes obvious from (13) and (16)-(18) that axial forces perform power on strain rates stemming from virtual translational motions of the cross sections in the tangential direction, while the bending moments perform power on strain rates stemming from virtual rotational motions around the gravitational centers of the generator lines.

Virtual power of external forces-ground pressure and impost forces
In tunnel shells, the external forces due to dead load (gravitational forces) are typically negligible with respect to the tractions forces stemming from the action of the surrounding ground, at radial coordinate r = R + h/2, so that |T| h |f| in expression (1) for the virtual power of the external forces. Furthermore, relevant traction forces occur only at surfaces with normals +e r (ϕ), −e ϕ (ϕ b ), and +e ϕ (ϕ e ). Considering these specifications, the insertion of the shell-specific virtual velocities (10) into expression (1) for the external forces acting in an arch-like tunnel cross section yields Considering h R, (19) can be simplified to whereby we introduced the tunnel engineering-specific notions of ground pressure, reading as and axial forces N p,b = N p (ϕ b ) and N p,e = N p (ϕ e ) at the beginning and at the end of the tunnel segment.
Resorting to architectural terms, the latter may be called impost forces. Defining them, somewhat analogously to the ground pressure (21), as compressive forces, they read mathematically as and whereby use of (17) and (6) with n = −e x and n = e x , respectively, was made.

Principle of virtual power-differential equilibrium conditions
In order to obtain differential equations quantifying the equilibrium in the tunnel shell segment, expression (4) for the principle of virtual power is considered to hold for the virtual velocity format (10) with arbitrary shell-specific virtual velocity fieldsv C r andv C ϕ . Therefore, Eq. (4) needs to be combined with the shell-specific expressions (16) and (20) for the power of the internal and the external forces, respectively; and the latter need to undergo partial integration. For the internal forces, we get whereby we considered vanishing bending moments at the surfaces with ϕ = ϕ b and ϕ = ϕ e , Insertion of (24) and (20) into Eq. (4), and requiring the result to hold for arbitrary values ofv C r andv C ϕ yields the differential equilibrium conditions valid along the entire arch segment, as well as the natural boundary conditions valid at the beginning and the end of the tunnel segment, 2.6 Ground pressure-induced distributions of internal axial forces and bending moments in tunnel shell segment As a first step to obtain mathematical solutions for the differential equations (26.1), (26.2) is differentiated with respect to ϕ, and the corresponding result is solved for This equation is then re-inserted into (26.1), yielding a differential equation for n(ϕ) only. The latter reads as This differential equation can be readily integrated for given functions quantifying the ground pressure distribution along the circumferential coordinate ϕ. In the following, we are interested in simple functional forms which eventually allow for conversion of displacement components from three geodetic measurement points into ground pressure values defined at four points along the circumferential coordinate ϕ. In more detail, we consider the weighted superposition of four cubic shape functions, the values of which are either one or zero at equi-distant points along the coordinateφ = ϕ − ϕ b . In the case of cubic polynomials, we have for the ground pressure Thereby, the cubic polynomials A i , i = 1, 2, 3, 4, read as with ϕ = ϕ e − ϕ b the opening angle of the tunnel shell. In (30), G p,1 is the pressure at positionφ = 0, G p,2 is the pressure at positionφ = ϕ/3, G p,3 is the pressure at positionφ = 2 ϕ/3, and G p,4 is the pressure at positionφ = ϕ.
The solution of the differential equation (29) for the ground pressure according to (30) and (31) yields where we considered Eqs. (22) and (23) to label the forces N p,b and N p,e pressing from the outside onto the imposts of the considered tunnel shell segment. The solution of the differential equation (26.1) for the ground pressure according to Eqs. (30) and (31), and for the axial forces according Eq. (32) yields whereby we considered vanishing bending moment at the beginning of the tunnel shell segment, see (25.1). Specification of (33) for m ϕ (ϕ e ) = m ϕ (φ = ϕ) = 0 yields the remarkable result In other words, independent of the actual ground pressure distribution, the impost forces at the beginning and the end of the tunnel segment are equal.

Strength, elasticity, and creep properties of concrete-Laplace Carson transform into inverse time domain
For the purposes of practical concrete engineering, it is customary to approximate the temporal evolution of the uniaxial strength of concrete under isothermal conditions at 20 centigrades by a suitable fitting function, such as the one given in [4], with the dimensionless strength evolution parameter s E , and with the time t being given in the unit of measurement "days" and being resolved down to tens of minutes. Typical values relevant for shotcrete tunneling are given in Table 1 and illustrated in Fig. 3a. Analogous relationships are employed for the elasticity development, see again Table 1 and Fig. 3b for typical values concerning shotcrete, and the parameter α amounts to one for quartz-and limestone-based concretes [1]. Both the uniaxial strength and Young's modulus are driven by the hydration reaction leading to changes in the material's microstructure, and this can be quantified by multiscale micromechanical models such as the one proposed in [18] and employed in [39]. In more detail, the uniaxial strength is governed by the hydration degree, the (initial) water-to-cement ratio, and the aggregate volume fraction in an RVE of concrete (see Appendix A), which allows one to reconstruct the evolution of the hydration degree associated with property developments (35) and (36) as see Fig. 4. The degree of hydration also governs the (ultra-)short term creep of concrete, as evidenced by three-minute creep tests [1,14,33], which give access to power-law type creep functions describing creep over time spans during which the hydration degree is virtually constant (i.e., from minutes in early-age concrete, to weeks in decade-old concrete [17]), with t being resolved down to the time regime of seconds, with the power-law exponent amounting typically to 0.25 [17], with the creep modulus E c and Young's modulus E depending on the degree of hydration ξ , and with t 0 = 1 d = 86 400 s as a reference time. E c is also available from isothermal creep tests performed at a temperature of 20 centigrades [1], namely in the format  Table 1 Accordingly, evaluation of (36) and (40) at the same time instants, and forming corresponding data pairs of hydration degree and (Young's or creep) modulus, gives access to the material functions E(ξ ), E c (ξ ), see Fig. 5. J (ξ, t) describes the behavior of a material sample with (constant) degree of hydration ξ , under uniaxial stress σ = σ ϕϕ e ϕ ⊗ e ϕ , according to a hereditary integral of the Boltzmann type [3,20], which, after Laplace-Carson transformation reads as with the transformed function J (ξ, p) exhibiting the format Lateral normal strains are customarily considered by a constant Poisson's ratio of ν = 0.2, in the form and in a fully 3D isotropic setting, (44) reads as ⎡ However, tunnel shells, as a rule, do not undergo uniaxial stresses; in fact, as a first approximation, they are rather characterized by ε zz ( p) ≈ 0. Corresponding specification of (47) and solution for the unknowns σ ϕϕ ( p) and σ zz ( p) yields We note that Eqs. (48) and (49) describe the stress reaction of a piece of shotcrete with hydration degree ξ , situated within the tunnel shell, to normal strain ε ϕϕ evolving with the inverse time variable p.

Creep upscaling from material to shell level-backtransformation into the time domain
In order to relate the stress resultants n ϕ and m ϕ , see Eqs. (17) and (18), to the strains they provoke, (48.1) is inserted into (17) and (18), respectively, yielding where we consider, for the sake of simplicity, homogeneous viscoelastic properties within the tunnel shell segment; these properties being still associated with a constant degree of hydration ξ . The tangential normal strains ε ϕϕ and the corresponding tangential and radial displacements are introduced in a linearized fashion, by temporal integration over constant strain rates which are formally identical with (13); this results straightforwardly in whereby the term proportional to (1/r ) is related to stretching, while the bending-related term is proportional to (r − R). Insertion of (51) into (50.1) yields whereby the last term holds for (h/R) 1, in particular because Insertion of (51) into (50.2) yields The last term in (54) holds for (h/R) 1, because of the following considerations: Firstly, the integrals occurring in (54) can be evaluated as follows: Secondly, the bending-related portions of the circumferential normal strains on the outer shell surfaces, where r = (R + h/2), need to be of the same size as the stretching-related portions of these strains; mathematically, this can be expressed, when considering (51), as Given in addition that (56) readily implies that so that only the terms d 2 u C r /dϕ 2 and du C ϕ /dϕ are not negligible in the expression for the bending moment according to the last line of (54).
As a first step to obtain mathematical solutions for the differential equations (52) and (54), (52) is solved for du C ϕ /dϕ, yielding This equation is then re-inserted into (54), yielding a differential equation for u C r only. The latter reads as Finally, an expression for the angular rotation angle θ C z along the circumferential coordinateφ is derived. Given the small actual deformations, this rotational angle can be approximated by temporal integration over constant angular velocities which are formally identical to the virtual angular velocities introduced in Eq. (9) and right above this equation. They fall into rotational portions associated with the origin and the shell center surface, reading asv C ϕ /R and (dv C r /dϕ)/R, so that the rotational angle of the shell generator line positioned at coordinate ϕ can be expressed in terms of radial and circumferential displacement components as The solution of the differential equations (59) and (60) for the axial forces (32) and the bending moments (33), as well of Eq. (61) yields an expression for the radial and circumferential displacements of the center surface and the rotational angle of the shell generator. The determination of the three integration constants is done with u C r,b = u C r (φ = 0) and u C ϕ,b = u C ϕ (φ = 0); which denotes the radial and circumferential displacements of the center surface at the beginning of the tunnel shell segment, and with θ C z,b = θ C z (φ = 0); which denotes the rotational angle of the shell generator line around an axis oriented in e z -direction and positioned in the shell center, at the beginning of the tunnel shell segment.
The solution of the differential equation (60) for the axial forces (32) and the bending moments (33), and with the considered integration constants yields We explicitly note the interesting structure of the force-driven portion of the solution for the radial displacements (62), which, when remembering (49.1), can be written as the product of the uniaxial creep function J (ξ, p) with the ground/impost pressure-weighted sum of time-invariant influence functions I, depending on ν, R, h, andφ; according to with the traction-to-displacement influence functions I given as Eqs. (99)-(103) in Appendix B. Backtransformation of (63) to the time domain according to The solution of the differential equation (59) for the axial forces (32) and the radial displacements (62), and with the considered integration constants yields In analogy to (63), this expression can be written in terms of polar angle-specific influence functions, resulting in the following relation after transformation into the time domain: with the influence functions I given as Eqs. (104)-(108) in Appendix B. The final mathematical solution for the rotational angle of the shell generator yields In analogy to (63), this expression can be written in terms of polar angle-specific influence functions, resulting in the following relation after transformation into the time domain: with the influence functions I given as Eqs. (109)-(113) in Appendix B.

Aging and nonlinear creep-degree of utilization
Equations (65), (67), and (69) quantify the creep behavior of the entire shotcrete tunnel shell, in terms of displacements and cross-sectional rotations as functions of ground and impost pressures, for a constant degree of hydration. However, the latter itself evolves with time as well. Hence, the aforementioned relations, strictly speaking, are only valid for a very short time interval during which the hydration degree is actually constant. Accordingly, adopting the experimentally validated conceptual reasoning outlined in [28], we apply these creep relations in rate form, for each and every time instant and the then prevailing degree of hydration. Differentiation of the corresponding parameter integrals results iṅ whereby we made use of J ξ(t), 0 = 1/E ξ(t) according to Eq. (39). In analogy to (70), the rate forms for the circumferential displacements and the rotational angles around e z read aṡ Finally, it is known that the creep compliance increases nonlinearly with the stress once a critical load level is exceeded. This is elegantly quantified in terms of the affinity concept of Ruiz et al [25], according to which the rate of the creep function (39) needs to be multiplied by a factor η, with the affinity factor reading as whereby L is the level of loading associated with the strength of the material. In the line of [38], L is associated with a Drucker-Prager strength criterion applied to the stresses at the center line of the tunnel shell, so that with parameters α DP and k DP being related to the uniaxial and biaxial compressive strengths f c and f b of shotcrete, where κ = 1.15 follows from standard tests [15], and f c follows the evolution given by Eq. (35). We note that the format of (74) maintains the consideration of homogeneous creep properties, as introduced in Eq. (50.2).

Geometrical and material properties
The analytical mechanics model for an aging viscoelastic cylindrical shell segment, as developed in Sects. 2 and 3, is eventually applied to a benchmark example in NATM tunneling, which has been analyzed by various types of "hybrid methods" combining geodetic measurements with material and structural mechanics tools [11,12,[36][37][38]: This benchmark example is built on cross section MC1452 of the Sieberg tunnel, a tunnel constructed in the 1990s as part of the high-speed railway line connecting Vienna and Salzburg. In terms of the geometrical properties introduced in Sects. 2 and 3, it is characterized by radius R = 6.20 m, a thickness h = 0.30 m, and an opening angle ϕ = 2.92 rad = 167.30 • , see Fig. 6. During the top heading excavation and installation stage, it was equipped with three optical reflectors delivering displacement vectors in three measurement points (MPs), MP1, MP2, and MP3, see Fig. 6; and we here consider the corresponding measurements over the first 28 days of the lifetime of the Sieberg tunnel, see Table 2. Moreover, we consider a typical shotcrete mixture with the cement type CEM II/A-S 42.5R, effective water-to-cement mass ratio (w/c) ef f = 0.5, aggregate-to-cement mass ratio a/c = 4.48, and volume fractions of aggregates f con agg = 0.70. For such a composition, Figures 13  and 3 of [38] suggest the following values for the degree of hydration and uniaxial strength after 28 days of shotcrete age: ξ = 0.8762 and f c,28d = 58.14 MPa.  Table 2 Radial and circumferential displacement components measured (in meters) at three geodetic reflectors installed within cross section MC1452 of the Sieberg tunnel; as seen in Fig. 6 viewing time The same temporal approximation type is used for the ground and impost pressure values N p and G p,i , with  Table 2 the result into (48.2) while considering (49.2), and defining the axial normal force by replacing, in (17), "ϕ" by "z", yields

Conclusions
The consideration of polar components of point-wise measured displacement vectors in combination with a tunnel-specific shell theory and the viscoelastic modeling of aging shotcrete provides analytical access to the ground pressure distribution along the tunnel circumference, impost forces, and all quantities arising from the action of the latter external forces: distributions of all internal normal forces and bending moments, of radial and circumferential displacements, and of the degree of utilization. The "white-box" nature of this approach, providing a closed-form expression for advanced mechanics-driven data evaluation, renders it, in the opinion of the authors, as a prime source for reliable and clear rule development in the ongoing discussion concerning artificial intelligence and "big data" in geotechnical engineering. In particular, the presented concept can be straightforwardly extended towards more complex geometries and construction sequences. This task is planned in close interaction with the tunnel engineering industry. At the same time, and from a more pragmatic perspective, our approach also provides novel insights which may support the calibration of state-of-the-art Finite Element models encompassing the tunnel shell and the bolt-reinforced rock [21].
Funding Open access funding provided by TU Wien Bibliothek.
Acknowledgements The authors gratefully acknowledge project FFG-COMET #882504 "Rail4Future: Resilient Digital Railway Systems to enhance performance".
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A Approximation of the concrete strength-based on a micro-to-macro validated engineering mechanics model
[18] provides a mathematical approximation for the uniaxial compressive strength of concrete, as a function of material composition and maturity, i.e., a combination of the effective water-to-cement mass ratios (w/c) ef f , the aggregate-to-cement mass ratio a/c, the volume fractions of aggregates f con agg and the hydration degree ξ . In more detail the uniaxial compressive strength of the hydrate phase results from the Mohr-Coulomb failure criterion as The uniaxial compressive strength of the cement paste can be expressed as with the aggregate volume fraction as function of the (initial) aggregate-to-cement ratio (a/c), (initial) waterto-cement ratio (w/c), and the mass densities of aggregates, clinker and water, ρ agg , ρ clin , and ρ H 2 O For a known shotcrete composition, the development over time of the uniaxial compressive strength of concrete can be described by (96), and simultaneously by means of the fib Model Code according to (35). Equating these two relations, while choosing specific model code quantities (namely, uniaxial compressive strength values reached after 28 days, which amount to of 50, 55, and 60 MPa; and a strength evolution parameter of s E = 0.18) and compositional characteristics ((effective) water-to-cement ratios (w/c) ef f ∈ {0.450, 0.475, 0.500, 0.525} and f con agg = 0.70 based on the concrete composition for the test campaign ULB I according to [18]), one arrives at evolutions of the degree of hydration as shown in Figs. 4 and 5.

B Time-invariant influence functions
The time-invariant influence functions I for the radial displacements can be expressed as The time-invariant influence functions I for the tangential displacements can be expressed as The time-invariant influence functions I for the cross-sectional rotation angle can be expressed as