From micron-sized needle-shaped hydrates to meter-sized shotcrete tunnel shells: micromechanical upscaling of stiffness and strength of hydrating shotcrete

Abstract

Knowledge on the stresses in shotcrete tunnel shells is of great importance, as to assess their safety against severe cracking or failure. Estimation of these stresses from 3D optical displacement measurements requires shotcrete material models, which may preferentially consider variations in the water–cement and aggregate–cement ratios. Therefore, we employ two representative volume elements within a continuum micromechanics framework: the first one relates to cement paste (with a spherical material phase representing cement clinker grains, needle-shaped hydrate phases with isotropically distributed spatial orientations, a spherical water phase, and a spherical air phase; all being in mutual contact), and the second one relates to shotcrete (with phases representing cement paste and aggregates, whereby aggregate inclusions are embedded into a matrix made up by cement paste). Elasticity homogenization follows self-consistent schemes (at the cement paste level) and Mori–Tanaka estimates (at the shotcrete level), and stress peaks in the hydrates related to quasi-brittle material failure are estimated by second-order phase averages derived from the RVE-related elastic energy. The latter permits upscaling from the hydrate strength to the shotcrete strength. Experimental data from resonant frequency tests, ultrasonics tests, adiabatic tests, uniaxial compression tests, and nanoindentation tests suggest that shotcrete elasticity and strength can be reasonably predicted from mixture- and hydration-independent elastic properties of aggregates, clinker, hydrates, water, and air, and from strength properties of hydrates. At the structural level, the micromechanics model, when combined with 3D displacement measurements, predicts that a decrease of the water–cement ratio increases the safety of the shotcrete tunnel shell.

Appendices

Appendix 1: list of symbols and abbreviations

a/c :

aggregate–cement ratio

$\tilde{A}$ :

macroscopic chemical affinity

C-S-H:

abbreviation for calcium silicate hydrates

$\underline{\underline{\underline{\underline C}}}$ :

fourth-order stiffness tensor

$\underline{\underline{\underline{\underline C}}}_p$ :

stiffness tensor of material phase p

$\underline{\underline{\underline{\underline C}}}_{\rm agg}$ :

stiffness tensor of aggregates

$\underline{\underline{\underline{\underline C}}}_{\rm cp}$ :

stiffness tensor of cement paste

$\underline{\underline{\underline{\underline C}}}_{\rm sc}$ :

stiffness tensor of shotcrete

C _p,ijrs :

ijrs-th component of $\underline{\underline{\underline{\underline C}}}_p$

$\underline{\underline{\underline{\underline C}}}_{\rm sc,aniso}$ :

anisotropic (transversely-isotropic) stiffness tensor of shotcrete

d :

characteristic length of inhomogeneities within the RVE

d _cp :

characteristic length of inhomogeneities within the RVE_cp of cement paste

d _sc :

characteristic length of inhomogeneities within the RVE_sc of shotcrete

dV :

infinitesimal volume element

${{\mathcal{D}}}$ :

characteristic length of (tunnel) structure, built up of RVEs (of shotcrete)

$\underline{\underline{\underline{\underline D}}}_{\rm sc}$ :

compliance tensor of shotcrete

e _i :

base vectors of orthonormal base frames oriented along needle-shaped hydrates, i = I, II, III

e _j :

base vectors of orthonormal base frame within an RVE, j = x, y, z

E _a :

activation energy

$\underline{\underline E}$ :

macroscopic second-order strain tensor

$\underline{\underline E} _{\rm cp}$ :

macroscopic second-order strain tensor of cement paste

$\underline{\underline E} _{\rm sc}$ :

macroscopic second-order strain tensor of shotcrete

E :

Young’s modulus

E ^exp_sc :

experimentally obtained Young’s modulus of shotcrete

E ^mod_sc :

model-predicted Young’s modulus of shotcrete

${{\mathcal{E}}}$ :

relative error between model-predicted Young’s modulus E ^mod_sc and corresponding experimentally obtained value E ^exp_sc

$\bar{{\mathcal{E}}}$ :

mean relative error over several model-predicted and measured Young’s moduli

${{\mathcal{F}}}$ :

relative error between model-predicted uniaxial compressive strength $\Upsigma_{\rm mod}^{\rm ult}$ and corresponding experimentally obtained value $\Upsigma_{\rm exp}^{\rm ult}$

$\bar{{\mathcal{F}}}$ :

mean relative error over several model-predicted and measured uniaxial compressive strengths

f _air :

volume fraction of air within RVE_cp of cement paste

f _clin :

volume fraction of cement clinker within RVE_cp of cement paste

f _hyd :

volume fraction of hydration products within RVE_cp of cement paste

$f_{\rm H_2O}$ :

volume fraction of water within RVE_cp of cement paste

f _p :

volume fraction of material phase p

$\bar{f}_{\rm agg}$ :

volume fraction of aggregates within the RVE_sc of shotcrete

$\bar{f}_{\rm cp}$ :

volume fraction of cement paste within the RVE_sc of shotcrete

$\bar{f}_{{\rm hyd}}$ :

volume fraction of hydrates within the RVE_sc of shotcrete

h :

thickness of tunnel shell

$\underline{\underline{\underline{\underline I}}}$ :

symmetric fourth-order unity tensor

i :

index

$\underline{\underline{\underline{\underline J}}}$ :

volumetric part of $\underline{\underline{\underline{\underline I}}}$

j :

index

k :

parameter describing the Drucker–Prager failure surface (of shotcrete)

k _air :

bulk modulus of air

k _agg :

bulk modulus of aggregates

k _clin :

bulk modulus of cement clinker

k _hyd :

bulk modulus of hydration products

$k_{\rm H_2O}$ :

bulk modulus of water

k _p :

bulk modulus of material phase p

$\underline{\underline{\underline{\underline{K}}}}$ :

deviatoric part of $\underline{\underline{\underline{\underline I}}}$

ℓ:

characteristic length of RVE

ℓ_cp :

characteristic length of RVE_cp of cement paste

ℓ_sc :

characteristic length of RVE_sc of shotcrete

$\fancyscript{L}$ :

level of loading

$\overline{\fancyscript{L}}$ :

$\fancyscript{L}$ averaged over the tunnel shell thickness

N :

number of summation terms

$\underline{\underline{\underline{\underline{P}}}}_{p}^{0}$ :

fourth-order Hill tensor of phase p embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

$\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{0}$ :

fourth-order Hill tensor of a needle-shaped phase embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

$\underline{\underline{\underline{\underline P}}}_{\rm sph}^{0}$ :

fourth-order Hill tensor of a spherical phase embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

p :

index denoting material phases clinker, hydrates, water, and air

q :

index denoting material phases clinker, water, and air

R :

universal gas constant, R = 8.31447 J mol⁻¹ K⁻¹

RVE:

abbreviation for representative volume element

RVE_cp :

abbreviation for a representative volume element of cement paste

RVE_sc :

abbreviation for a representative volume element of shotcrete

r :

index

s :

index

$\underline{\underline{\underline{\underline S}}}_{p}^{0}$ :

fourth-order Eshelby tensor of phase p embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

$\underline{\underline{\underline{\underline S}}}_{\rm cyl}^{0}$ :

fourth-order Eshelby tensor of a needle-shaped phase embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

$\underline{\underline{\underline{\underline S}}}_{\rm sph}^{0}$ :

fourth-order Eshelby tensor of a spherical phase embedded in a matrix with stiffness tensor $\underline{\underline{\underline{\underline C}}}^0$

t :

time variable

T :

absolute temperature

v _sc :

velocity of longitudinal wave propagation in shotcrete

W :

elastic energy stored in the RVE

w/c :

water–cement ratio

${\underline{x}}$ :

position vector labeling locations within the RVE and on its boundary

α :

parameter describing the Drucker–Prager failure surface (of shotcrete)

α ⁰ :

volumetric part of $\underline{\underline{\underline{\underline S}}}_{\rm sph}^{0}$

β ⁰ :

deviatoric part of $\underline{\underline{\underline{\underline S}}}_{\rm sph}^{0}$

${\underline{\underline{\gamma}}}$ :

inverse of the acoustic tensor

$\underline{\underline{\underline{\underline \Gamma}}}$ :

fourth-order tensor needed for determination of the Hill tensor for needle-shaped phases embedded in a transversely isotropic matrix

δ _ij :

Kronecker delta

∂Ω:

surface of the RVE

∂Ω_cp :

surface of the RVE_cp of cement paste

${\underline{\underline\varepsilon}}_p$ :

linearized second-order strain tensor of material phase p

${\underline{\underline{\varepsilon}}}^{\rm dev}$ :

deviatoric part of ${\underline{\underline\varepsilon}}$

$\overline{\overline{{\varepsilon}^{\rm dev}_p}}$ :

quadratic average of ${{\underline{\underline\varepsilon}}}^{\rm dev}$ over material phase p

κ:

factor relating the uniaxial to the biaxial compressive strength (of shotcrete)

$\underline{\underline{\kappa}}$ :

acoustic tensor

ϑ:

Euler angle (spherical coordinate)

μ _air :

shear modulus of air

μ _agg :

shear modulus of aggregates

μ _clin :

shear modulus of cement clinker

μ _hyd :

shear modulus of hydration products

$\mu_{\rm H_2O}$ :

shear modulus of water

μ _p :

shear modulus of material phase p

ν :

Poisson’s ratio

ρ _air :

mass density of air

ρ _agg :

mass density of aggregates

ρ _clin :

mass density of cement clinker

ρ _cp :

macroscopic mass density of cement paste

ρ _hyd :

mass density of hydration products

ρ _sc :

macroscopic mass density of shotcrete

$\rho_{\rm H_2O}$ :

mass density of water

$\underline{\underline{{\sigma}}}_{p}$ :

second-order stress tensor of material phase p

$\underline{\underline{\sigma}}_{p}^{\rm dev}$ :

deviatoric part $\underline{\underline{\sigma}}_{p}$

σ ^dev _p :

norm of stress deviator $\underline{\underline{\sigma}}_{p}^{\rm dev}$

$\overline{\overline{\sigma^{\rm dev}_{p}}}$ :

quadratic average of $\underline{\underline\sigma}_p^{\rm dev}$

$\sigma_{{\mathcal{E}}}$ :

standard deviation of ${{\mathcal{E}}}$

$\sigma_{{\mathcal{F}}}$ :

standard deviation of ${{\mathcal{F}}}$

$\underline{\underline\Upsigma}$ :

macroscopic second-order stress tensor

$\underline{\underline\Upsigma}_{\rm cp}$ :

macroscopic second-order stress tensor of cement paste

$\underline{\underline\Upsigma}_{\rm sc}$ :

macroscopic second-order stress tensor of shotcrete

$\Upsigma_{\rm exp}^{\rm ult}$ :

experimentally obtained uniaxial compressive strength of shotcrete

$\Upsigma_{\rm mod}^{\rm ult}$ :

model-predicted uniaxial compressive strength of shotcrete

$\Upsigma_{\rm sc,11}^{\rm comp,ult}$ :

model-predicted uniaxial compressive strength of shotcrete

φ:

Euler angle (spherical coordinate) describing hydrate orientation

ϕ :

cylindrical coordinate describing position in tunnel shell

ω _r :

flexural resonant frequency

Ω:

volume of the RVE

Ω_cp :

volume of RVE of cement paste

Ω_hyd :

subvolume of Ω_cp or of Ω_sc, occupied by hydrates

Ω _p :

subvolume of Ω_cp or of Ω_sc, occupied by material phase p

Ω_sc :

volume of RVE of shotcrete

ξ:

degree of hydration

${\underline{\xi}}$ :

displacement vector

${\underline{\zeta}}$ :

unit length vector indicating the surface of a unit sphere

·:

first-order tensor contraction

::

second-order tensor contraction

×:

vector cross product

⊗:

dyadic product

Appendix 2: vector/matrix notation for symmetric second-order and fourth-order tensors

For computationally economical realization of the tensor operations throughout this paper, second-order and fourth-order tensors are expressed through a compressed symmetric vector/matrix notation with normalized tensorial basis, often referred to as the Kelvin or the Mandel notation [12, 23, 33]: accordingly, a symmetric second-order tensor $\underline{\underline{A}}=A_{ij} \underline{e}_i\otimes\underline{e}_j$ (sum over indexes appearing twice: i, j = 1, 2, 3) is represented by a [6 × 1]-vector A of the form

$${{\mathbf{A}}}=\left[ \begin{array}{*{20}l} A_{11}&A_{22}&A_{33}&\sqrt{2}A_{23}&\sqrt{2}A_{13}&\sqrt{2}A_{12}\\ \end{array} \right]^{\rm T} $$

(49)

with A _ij , i,j ∈ [1,2,3], as the components of $\underline{\underline{A}}.$ A fourth-order tensor $\underline{\underline{\underline{\underline{B}}}}=B_{ijrs} \underline{e}_i\otimes\underline{e}_j\otimes\underline{e}_r\otimes \underline{e}_s$ (sum over indexes appearing twice: i, j, r, s = 1, 2, 3), with symmetries B _ijrs  = B _jirs  = B _ijsr , is represented by a [6 × 6]-matrix B, such that

$${{\mathbf{B}}}=\left[ \begin{array}{*{20}l} B_{1111} & B_{1122} & B_{1133} & \sqrt{2}B_{1123} & \sqrt{2}B_{1113} & \sqrt{2}B_{1112} \\ B_{2211} & B_{2222} & B_{2233} & \sqrt{2}B_{2223} & \sqrt{2}B_{2213} & \sqrt{2}B_{2212} \\ B_{3311} & B_{3322} & B_{3333} & \sqrt{2}B_{3323} & \sqrt{2}B_{3313} & \sqrt{2}B_{3312} \\ \sqrt{2}B_{2311} & \sqrt{2}B_{2322} & \sqrt{2}B_{2333} & 2B_{2323} & 2B_{2313} & 2B_{2312} \\ \sqrt{2}B_{1311} & \sqrt{2}B_{1322} & \sqrt{2}B_{1333} & 2B_{1323} & 2B_{1313} & 2B_{1312} \\ \sqrt{2}B_{1211} & \sqrt{2}B_{1222} & \sqrt{2}B_{1233} & 2B_{1223} & 2B_{1213} & 2B_{1212} \\ \end{array} \right]$$

(50)

with B _ijrs , i,j,r,s∈[1,2,3], as the components of $\underline{\underline{\underline{\underline{B}}}}.$

Appendix 3: computational realization of the strength criterion

Herein, we describe a strategy to compute the term within the curly brackets in the elastic limit criterion (35), i.e., the expression

$$ \max\limits_{\varphi,\vartheta}\left[\lim\limits_{\Updelta\varphi, \Updelta\vartheta\rightarrow 0}\left(\frac{{\mu_{{\rm hyd}}^2}} {\bar{f}_{{\rm hyd};\varphi,\vartheta}}(\underline{e}_z\otimes\underline{e}_z): \underline{\underline{\underline{\underline{C}}}}_{\rm sc}^{-1}: {\frac{{\partial\underline{\underline{\underline{\underline{C}}}}_{\rm sc}}} {\partial \mu_{{\rm hyd};\varphi,\vartheta}}}: \underline{\underline{\underline{\underline {C}}}}_{\rm sc}^{-1}:(\underline{e}_z\otimes\underline{e}_z)\right)^ {\frac{1}{2}}\right] $$

(51)

The required derivative will be approximated by a finite difference quotient. To minimize the efforts to check all hydrate orientations, the derivative of (51) is computed only for needle-shaped hydrates pointing in z-direction (ϑ = 0). Consistently, the macroscopic loading direction (originally ${\underline{e}}_z)$ is (formally) rotated in all directions of ${\mathbb{R}}^3\!\!:$

$$\begin{aligned} &(51)=\max\limits_{{\varphi},{\vartheta}}\left\{\left[\left(\underline{e}_r \left(\varphi,\vartheta\right)\otimes\underline{e}_r \left(\varphi,\vartheta\right)\right):\underline{\underline{\underline{\underline{C}}}}_{\rm sc}^{-1}\right.\right.\\ &\quad \left.\left. : \lim\limits_{\Updelta\varphi,\Updelta\vartheta\rightarrow0}\left({\frac{\mu_{{\rm hyd}}^2} {\bar{f}_{{\rm hyd};\varphi,\vartheta=0}}}{\frac{\partial \underline{\underline{\underline{\underline{C}}}}_{\rm sc}} {\partial\mu_{{\rm hyd};\varphi,\vartheta=0}}}\right):\underline{\underline{\underline{\underline{C}}}}_{\rm sc}^{-1}:\left(\underline{e}_r\left(\varphi,\vartheta\right)\otimes \underline{e}_r\left(\varphi,\vartheta\right)\right)\right]^{\frac{1} {2}}\right\} , \end{aligned}$$

(52)

with

$$ \underline{e}_r=\cos\varphi \sin\vartheta \underline{e}_x+\sin\varphi \sin\vartheta \underline{e}_y+\cos\vartheta \underline{e}_z. $$

(53)

The term in (52), involving the limits Δφ→0 and Δϑ→0 as well as the derivative of the homogenized stiffness of shotcrete with respect to the shear modulus of needle-shaped hydrates pointing in z-direction, is approximated as

$$ \lim\limits_{\Updelta\varphi,\Updelta\vartheta\rightarrow 0}\left(\frac{\mu_{{\rm hyd}}^2} {\bar{f}_{{\rm hyd};\varphi,\vartheta=0}} \frac{\partial \underline{\underline{\underline{\underline{C}}}}_{\rm sc}} {\partial\mu_{{\rm hyd};\varphi,\vartheta=0}}\right) \approx{\frac{\mu_{{\rm hyd}}^2} {\bar{f}_{{\rm hyd};\varphi,\vartheta=0}}} \left[{\frac{\underline{\underline{\underline {\underline{C}}}}_{\rm sc,aniso}{\left(\mu_{{\rm hyd},\varphi,\vartheta=0}= 1.005 \mu_{{\rm hyd}}\right)}-\underline{\underline{\underline{\underline {C}}}}_{\rm sc}} {0.005 \mu_{{\rm hyd}}}}\right] , $$

(54)

with

$$ \bar{f}_{{\rm hyd};\varphi,\vartheta=0}=0.001 . $$

(55)

The chosen values of one per mille volume fraction of $\underline{e}_z$-oriented hydrates [see (55)] and of five per mille increase of the corresponding hydrate shear modulus [see (54)] are (a) small enough as to ensure that the finite difference quotient accurately approximates the expression on the left-hand side of (54), and (b) large enough as to avoid numerical problems related to the computation of the finite difference quotient. $\underline{\underline{\underline{\underline C}}}_{\rm sc,aniso}$ is the Mori–Tanaka stiffness estimate for shotcrete considering that the shear modulus of the hydrates pointing in z-direction is by five-tenth of a percent larger than the one of all other hydrates. Hence, $\underline{\underline{\underline{\underline C}}}_{\rm sc,aniso}$ is a transversely isotropic tensor. Its computation requires the following modification of (23):

$$ \begin{aligned} \underline{\underline{\underline{\underline C}}}_{\rm sc,aniso}&=\left\{\bar{f}_{\rm cp} \underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}+\bar{f}_{\rm agg} \underline{\underline{\underline{\underline{C}}}}_{\rm agg}:\left[\underline{\underline{\underline{\underline{I}}}}+\underline{\underline{\underline{\underline P}}}_{\rm sph}^{\rm cp}:(\underline{\underline{\underline{\underline{C}}}}_{\rm agg}- \underline{\underline{\underline{\underline{C}}}}_{\rm cp,aniso})\right]^{-1}\right\}\\ &\quad :\left\{\bar{f}_{\rm cp} \underline{\underline{\underline{\underline{I}}}}+\bar{f}_{\rm agg}\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline{P}}}}_{\rm sph}^{\rm cp}:(\underline{\underline{\underline{\underline{C}}}}_{\rm agg}-\underline{\underline{\underline{\underline{C}}}}_{\rm cp,aniso})\right]^{-1}\right\}^{-1} .\end{aligned}$$

(56)

In (56), $\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}$ is the self-consistent stiffness estimate of cement paste considering that the shear modulus of the hydrates pointing in z-direction is by five-tenth of a percent larger than the one of the other hydrates. Hence, $\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}$ is a transversely isotropic tensor, the computation of which requires the following modification of (19)–(21):

$$ \begin{aligned} \underline{\underline{\underline{\underline{C}}}}_{\rm cp,aniso}&= \left\{\sum_{ \begin{array}{*{20}l} p={\rm clin},\\ {\rm H_2{\rm O,air}}\\ \end{array}} f_{p} \underline{\underline{\underline{\underline C}}}_{p }:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm sph}^{\rm cp}:(\underline{\underline{\underline{\underline C}}}_{p }-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso})\right]^{-1}\right.\\ &\left.\quad +\sum_{{i}=1}^{15} \frac{f_{{\rm hyd}}}{15} \underline{\underline{\underline{\underline C}}}_{{\rm hyd}}:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\varphi_{i },\vartheta_{i }):(\underline{\underline{\underline{\underline C}}}_{{\rm hyd}}-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}) \right]^{-1}\right.\\ &\left.\quad - \bar{f}_{{\rm hyd};\varphi,\vartheta=0} \underline{\underline{\underline{\underline C}}}_{{\rm hyd}}:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\vartheta=0): (\underline{\underline{\underline{\underline C}}}_{{\rm hyd}}-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}) \right]^{-1}\right.\\ &\left.\quad + \bar{f}_{{\rm hyd};\varphi,\vartheta=0} \underline{\underline{\underline{\underline C}}}_{{\rm hyd},\vartheta=0}:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\vartheta=0): (\underline{\underline{\underline{\underline C}}}_{{\rm hyd},\vartheta=0}-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}) \right]^{-1}\right\}\\ &\quad :\left\{\sum_{ \begin{array}{*{20}l} q={\rm clin},\\ {\rm H_2O,air}\\ \end{array}} f_q\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm sph}^{\rm cp}:(\underline{\underline{\underline{\underline C}}}_q-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso})\right]^{-1}\right.\\ &\left.\quad + \sum_{j=1}^{15}\frac{f_{{\rm hyd}}}{15} \underline{\underline{\underline{\underline C}}}_{{\rm hyd}}:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\varphi_j,\vartheta_j): (\underline{\underline{\underline{\underline{C}}}}_{{\rm hyd}}- \underline{\underline{\underline{\underline{C}}}}_{\rm cp,aniso})\right]^{- 1} \right.\\ &\left.\quad - \bar{f}_{{\rm hyd};\varphi,\vartheta=0} \underline{\underline{\underline{\underline C}}}_{{\rm hyd}}:\left[\underline{\underline{\underline{\underline I}}}+\underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\vartheta\!=\!0)\!: \!(\underline{\underline{\underline{\underline C}}}_{{\rm hyd}}\!-\!\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}) \right]^{-1}\right.\\ &\left.\quad + \bar{f}_{{\rm hyd};\varphi,\vartheta=0} \underline{\underline{\underline{\underline C}}}_{{\rm hyd},\vartheta=0}:\left[\underline{\underline{\underline{\underline I}}}+ \underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}(\vartheta=0): (\underline{\underline{\underline{\underline C}}}_{{\rm hyd},\vartheta=0}-\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}) \right]^{-1}\right\}^{-1} \\ \end{aligned} $$

(57)

with

$$ \underline{\underline{\underline{\underline C}}}_{{\rm hyd}}=3 k_{{\rm hyd}} \underline{\underline{\underline{\underline{J}}}}+2 \mu_{{\rm hyd}} \underline{\underline{\underline{\underline{K}}}}\qquad \hbox{and} \qquad\underline{\underline{\underline{\underline C}}}_{{\rm hyd},\vartheta=0}= 3 k_{{\rm hyd}} \underline{\underline{\underline{\underline{J}}}}+2 (1.005 \mu_{{\rm hyd}}) \underline{\underline{\underline{\underline{K}}}} . $$

(58)

The terms in the second and the sixth line of (57) refer to integrals over all hydrate orientations, where the intrinsic elastic stiffness of hydrates (Table 1) is considered. The terms in the third, fourth, seventh, and eighth line of (57) refer to a bundle of hydrates pointing in z-direction, with volume fraction $\bar{f}_{{\rm hyd};\varphi,\vartheta=0}$ [see (55)]. The terms in the third and the seventh line of (57) subtract (from the aforementioned integrals over all hydrate orientations) the contribution of such a bundle of hydrates with the intrinsic shear modulus μ _hyd of Table 1, while the terms in the fourth and the eighth line of (57) add—as required for computation of the finite difference quotient (54)—the contribution of such a bundle of hydrates with the increased shear modulus 1.005μ _hyd. Notably, in both (56) and (57), all Hill tensors need to account for the anisotropy of $\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}$. Differing from the expressions (13)–(17), such Hill tensors are given in the subsequent Appendix 4.

Appendix 4: computation of Hill tensors

Nonzero components of the Hill tensor for spherical inclusions (clinker, water, and air) in the transversely isotropic cement paste matrix of stiffness $\underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}$ [see (57)], $\underline{\underline{\underline{\underline P}}}^{\rm cp}_{\rm sph},$ read as [21, 25]

$$ \begin{aligned} P_{\rm sph,1111}&=\frac{1} {16} \int\limits_{-1}^1-(-5C_{1111}x^4C_{3333}-3C_{1122}x^2C_{3333}- 3C_{1122}x^4C_{2323}\\ &\quad +3C_{1122} x^4C_{3333}+5C_{1111}x^4C_{2323}- 10C_{1111}C_{2323}x^2+2x^4C^{2}_{1133}\\ &\quad +8C_{2323}x^4C_{3333}- 6C^{2}_{2323}x^4+4C_{2323}x^4C_{1133}+6C_{1122}C_{2323}x^2\\ &\quad +5C_{1111}C_{2323}+5C_{1111}x^2C_{3333}- 4C_{2323}x^2C_{1133}+6C^{2}_{2323}x^2\\ &\quad -2x^2C^{2}_{1133}-3C_{1122}C_{2323})(-1+x^2)/{{\mathcal{D}}}_1{\rm d}x \\ \end{aligned} $$

(59)

$$ \begin{aligned} P_{\rm sph,1122}&=P_{\rm sph,2211}=\frac{1} {16} \int\limits_{-1}^1(C_{1111}C_{2323}- 2C_{1111}C_{2323}x^2+C_{1111}x^2C_{3333}\\ &\quad +C_{1122}C_{2323}- 2C_{1122}C_{2323}x^2+C_{1122}x^2C_{3333}+C_{1111}x^4C_{2323}\\ &\quad -C_{1111}x^4C_{3333}+C_{1122}x^4C_{2323}-C_{1122}x^4C_{3333}- 2C^{2}_{2323}x^2+2C^{2}_{2323}x^4\\ &\quad -4C_{2323}x^2C_{1133}+4C_{2323}x^4C_{1133}-2x^2C^{2}_{1133}\\ &\quad +2x^4C^{2}_{1133})(-1+x^2)/{{\mathcal{D}}}_1{\rm d}x \\ \end{aligned} $$

(60)

$$ P_{\rm sph,1133}=P_{\rm sph,3311}=\frac{1}{4} \int\limits_{-1}^1(-1+x^2)x^2(C_{2323}+C_{1133})/ {{\mathcal{D}}}_2 {\rm d}x $$

(61)

$$ \begin{aligned} P_{\rm sph,2323}&={\frac{1}{16}} \int\limits_{-1}^1(4C_{1111}C_{2323}x^2-8C_{2323}x^4C_{1133}-2x^4C^{2}_{1133}- C_{1122}x^4C_{3333}\\ &\quad-8C_{1111}x^4C_{2323}+3C_{1111}x^4C_{3333}+4C_{1111}x^4C_{1133}- 4C_{1122}x^4C_{1133}\\ &\quad+2C_{1122}x^6C_{1133}-2C_{1111}x^6C_{1133}+C_{1122}x^6C_{1111}- 3C_{1122}x^4C_{1111}\\ &\quad +3C_{1122}C_{1111}x^2- 2C_{1111}x^2C_{1133}+2C_{1122}x^2C_{1133}+8x^6C_{2323}C_{1133}\\ &\quad - 3x^6C_{1111}C_{3333}+4x^6C_{2323}C_{3333}+4C_{1111}x^6C_{2323}+C_{1122}x ^6C_{3333}\\ &\quad +3C^{2}_{1111}x^4-C^{2}_{1111}x^6+2C^{2}_{1133}x^6- 3C^{2}_{1111}x^2+C^{2}_{1111}\\ &\quad -C_{1122}C_{1111})/{{\mathcal{D}}}_1{\rm d}x \\ \end{aligned} $$

(62)

$$ P_{\rm sph,3333}=\frac{1}{2} \int\limits_{-1}^1x^2(x^2C_{2323}-C_{1111}x^2+C_{1111})/{{\mathcal{D}}}_2{\rm d}x $$

(63)

whereby

$$ \begin{aligned} {{\mathcal{D}}}_1 &= -2C^{2}_{1111}x^4C_{3333}+2C^{2}_{2323}x^6C_{3333}- 4C_{1111}C^{2}_{2323}x^4-3C^{2}_{1111}C_{2323}x^2\\ &\quad+C^{2}_{1111}x^2C_{3333}+2C_{1111}C^{2}_{2323}x^2- 2C_{2323}x^4C^{2}_{1133}-C_{1111}C^{2}_{1133}x^6\\ &\quad +2C_{1111}C^{2}_{1133}x^4+4C^{2}_{2323}x^6C_{1133}- 2C_{1122}C^{2}_{1133}x^4+2C_{2323}x^6C^{2}_{1133}\\ &\quad +3C^{2}_{1111}x^4C_{2323}+C_{1122}C^{2}_{1133}x^6- C^{2}_{1111}x^6C_{2323}+2C_{1111}x^6C^{2}_{2323}\\ &\quad +C^{2}_{1111}x^6C_{3333}-C_{1111}C^{2}_{1133}x^2- 4C^{2}_{2323}x^4C_{1133}+C_{1122}C^{2}_{1133}x^2\\ &\quad +C^{2}_{1111}C_{2323}-C_{1122}C_{1111}C_{2323}- C_{1122}x^6C_{1111}C_{3333}+4C_{1111}x^4C_{2323}C_{1133}\\ &\quad -2C_{1111}x^2C_{2323}C_{1133}- 4C_{1122}x^4C_{2323}C_{1133}+2C_{1122}x^2C_{2323}C_{1133}\\ &\quad +2C_{1122}x^6C_{2323}C_{1133}-2C_{1111}x^6C_{2323}C_{1133}- 3C_{1111}x^6C_{2323}C_{3333}\\ &\quad +2C_{1122}C_{1111}x^4C_{3333}-C_{1122}C_{2323}x^4C_{3333}- 3C_{1122}C_{1111}x^4C_{2323}\\ & \quad - C_{1122}C_{1111}x^2C_{3333}+3C_{1122}C_{1111}C_{2323}x^2+3C_{1111}C_{232 3}x^4C_{3333}\\ &\quad +C_{1122}x^6C_{1111}C_{2323}+C_{1122}x^6C_{2323}C_{3333}\\ \end{aligned} $$

(64)

and

$$ \begin{aligned} {{\mathcal{D}}}_2 &= 2C_{2323}x^4C_{1133}+C_{2323}x^4C_{3333}+C_{1111}x^4C_{2323}- 2C_{2323}x^2C_{1133}\\ &\quad -2C_{1111}C_{2323}x^2+C_{1111}C_{2323}+x^4C^{2}_{1133}- C_{1111}x^4C_{3333}-x^2C^{2}_{1133}\\ &\quad +C_{1111}x^2C_{3333}\\ \end{aligned} $$

(65)

whereby C _ijrs stands for C _{cp,aniso,ijrs} in (59)–(61). The integral expressions in (59)–(61) are evaluated numerically, based on an adaptive Simpson quadrature.

The Hill tensor for the needle-shaped hydrate phase, embedded in the transversely isotropic cement paste matrix, $\underline{\underline{\underline{\underline P}}}^{\rm cp}_{\rm cyl},$ reads as [19]

$$ \underline{\underline{\underline{\underline P}}}_{\rm cyl}^{\rm cp}=\frac{1} {2\pi} \int\limits_{0}^{2\pi}\underline{\underline{\underline{\underline \Gamma}}}(\varphi,\vartheta=\frac{\pi}{2}){\rm d}\varphi $$

(66)

with the fourth-order tensor $\underline{\underline{\underline{\underline {\Gamma}}}}$ reading as

$$ \begin{aligned} \underline{\underline{\underline{\underline{\Gamma}}}}&= \underline{\zeta} \mathop \otimes \limits^s \underline{\underline{\kappa}}^{-1} \mathop \otimes \limits^s {\underline{\zeta}}={\underline{\zeta}} \mathop \otimes \limits^s {\underline{\underline{\gamma}}} \mathop \otimes \limits^s {\underline{\zeta}},\\ {\Gamma}_{ijrs}&=\frac{1} {4}\left(\zeta_i{\gamma}_{jr}\zeta_s+\zeta_j{\gamma}_{ir}\zeta_s+\zeta_i {\gamma} _{js}\zeta_r+\zeta_j{\gamma}_{is}\zeta_r\right) \\ \end{aligned} $$

(67)

with the symmetrized dyadic product $\mathop \otimes \limits^s ,$ and with $\underline{\underline{\gamma}}$ as the inverse of the acoustic tensor $\underline{\underline{\kappa}}$ reading as

$$ \begin{aligned} \underline{\underline{\kappa}}&={\underline{\zeta}} \cdot \underline{\underline{\underline{\underline C}}}_{\rm cp,aniso}\cdot\underline{\zeta}\\ \kappa_{ij}&=\zeta_{r } C_{{\rm cp,aniso,}ijrs}\zeta_s \\ \end{aligned} $$

(68)

with the unit length vector $\underline{\zeta}$ indicating the surface of a unit sphere,

$$\underline{\zeta}=\sin\vartheta\cos\varphi\underline{e}_I+ \sin\vartheta\sin\varphi\underline{e}_{II}+\cos\vartheta\underline{e}_{I II} . $$

(69)

Appendix 5: tests of Lafarge [45]: determination of uniaxial compressive strength as a function of hydration degree

Since the shotcrete cured under quasiadiabatic conditions hydrated significantly faster than the one stored under quasi-isothermal conditions, Young’s moduli and compressive strength values measured at the same shotcrete age correspond to different hydration degrees, and consequently, they cannot be correlated. Instead, we combined the measured Young’s modulus–time relationship with the validated shotcrete elasticity model (providing a relation between Young’s modulus and hydration degree). This allowed for identifying the temporal evolution of the hydration degree of shotcrete hydrating under quasi-isothermal conditions. The obtained hydration degree–time relationship, $\xi=\xi(t)\rightarrow\dot{\xi}=\dot{\xi}(t),$ as well as the quasi-isothermal curing temperature T = 289.5 K, were the basis for identification of the normalized chemical affinity $\tilde{A},$ quantifying the hydration kinetics, $\tilde{A}(\xi(t))=\dot{\xi}(t)\exp(E_{\rm a}/(RT))$. ξ(t) and $\tilde{A}(\xi(t))$ give access to the affinity as a function of the hydration degree, as described by Ulm and Coussy [80] and Hellmich et al. [32]. Since $\tilde{A}(\xi)$ is an intrinsic shotcrete property depending on shotcrete composition only, the identified $\tilde{A}(\xi)$ relationship also applies for the same shotcrete exposed to quasiadiabatic conditions. The temporal evolution of the hydration degree in the latter shotcrete was evaluated by inserting both (a) the obtained $\tilde{A}(\xi)$ relationship and (b) the available temperature history of the quasiadiabatically hydrating shotcrete into Eq. 41. From the obtained hydration degree–time relationship, we read off the hydration degrees at the time instants of the strength measurements; corresponding pairs of strength values and hydration degrees are depicted through the squares in Fig. 5b.