In the multi-field model, mechanical, thermal, and hygral equilibrium is obtained simultaneously in a monolithic way. Since in this approach the variables are not uniformly distributed throughout a specimen, coupled three-dimensional finite element analyses of the considered specimens for determining the displacements, fluid pressures, and temperature have to be performed. For the latter, one-eighth of a cubic specimen is meshed, exploiting symmetry, using 512 elements with quadratic shape functions for displacements and temperature and linear shape functions for capillary pressure. A higher resolution is chosen near the drying surfaces. Thermal and hygral boundary conditions are governed by the ambient temperature and the ambient relative humidity . The convective heat transfer coefficient is denoted as and the convective mass transfer coefficient is named . In the following figures, the results at the center of the specimens are shown.
3.2.1. Comparison of Model Response with Experimental Results by Huber
The experimental data by Huber [15
] is chosen for comparing the predicted and measured evolution of the Young’s modulus and the uniaxial compressive strength. The respective experimental data is characterized by a good documentation and a comparatively small scatter of experimental data. In total, five test series, conducted on specimens with dimensions of
, are reported. Test series 5 is chosen for the calibration of the shotcrete models, and the numerical results, computed on the basis of those parameters, are compared to the experimental results of the test series 1 to 4. The shotcrete composition, listed in Table 1
, is identical for each series.
The measured Young’s modulus at the age of one day in test series 5 is specified in [15
] as E(1)
= 24,780 MPa. Unfortunately, no values for the Young’s modulus at the age of 28 days E(28)
are reported for any of the test series. For this reason, E(28)
is estimated as E(28)
= 30,000 MPa. Parameters
of the Meschke model, which govern the early age evolution of stiffness, are identified as
The viscoelastic compliance parameters
of the SCDP model are computed according to the estimation procedure [25
. To recover
30,000 MPa at the age of 28 days,
is determined as
by computing the effective Young’s modulus based on a duration of
. Flow compliance parameter
is not required in the present context.
The simulations using the Gawin model are based on the reported ambient temperature of 23
and ambient relative humidity of 50%. The heat and mass transfer coefficients are
. Thermal conductivity, the densities of solid, air, and water as well as their heat capacities are set to match typical values for ordinary concrete. An intrinsic permeability of 3 × 10−19
is used, which is considered to be a reasonable guess since it is in the range of values presented in [9
] for concrete. The porosity in the fully matured state is assumed to be 12%. The shotcrete specific values calibrated from test series 5 are the parameters governing hydration, the evolution of uniaxial compressive strength and effective Young’s modulus. The values are
= 0.03836 s
. The effective Young’s modulus at the center of the specimen is measured for the same time period
as for the SCDP model.
a shows the results of the calibration of the evolution laws for the Young’s modulus based on test series 5 of the experimental data in [15
] and Figure 2
b contains the validation of the respective evolution laws by means of test series 1 to 4 in [15
It can be seen that prior to the shotcrete age of 24 h, the evolution of the Young’s modulus, predicted by the Meschke model and the SCDP model, is closer to the experimental data than the one of the Schädlich model. This is mainly due to the considered delayed start of the hydration by both models. Although being based on several assumptions on the hygral and thermal parameters, the predictions using the Gawin model are of a similar quality as the SCDP model and the Meschke model.
The uniaxial compressive strength at the age of one day,
, measured in test series 5, is reported in [15
] as 18.2
. Again, as for the Young’s modulus, data for the uniaxial compressive strength at the age of 28 days
is not available. For this reason, a value of
is estimated. For the Gawin model,
a shows the results of the calibration of the evolution laws for the uniaxial compressive strength based on test series 5 of the experimental data in [15
] and Figure 3
b contains the validation of the respective evolution laws by means of test series 1 to 4 in [15
It can be seen that the compressive strength of shotcrete, younger than 24 h, is overestimated by the Schädlich model, while the Meschke model underestimates the strength between 8 and 24 h. The SCDP model is able to represent the delayed start of the evolution of material strength with respect to the casting time, which agrees well with experimental results. Again, the Gawin model and the SCDP model perform equally well. After the age of 24 h, all models predict similar compressive strengths.
3.2.2. Comparison of Model Response with Experimental Results by Müller
The experimental data by Müller [17
] is chosen for comparison of the shrinkage and creep behavior of the shotcrete models. Müller presented five series of experimental tests on shotcrete, consisting of four laboratory test series which are compared to one in situ test series. The shotcrete composition is listed in Table 2
Creep and shrinkage tests were carried out on unsealed specimens from steel molds with dimensions of . Ambient conditions, such as relative humidity or temperature, are not reported.
For the subsequently presented comparison, test series 3 and 4 are chosen because they are characterized by the smallest scatter of data. While test series 3 includes one creep test series, in test series 4, two different loading sequences were applied on two specimens each. The two test schemes are denoted as creep test series 4/1 and creep test series 4/2, which is consistent with the notation used in [17
]. The recorded strain from creep tests includes the combined autogenous and drying shrinkage strain as well as the basic and drying creep strain. Hence, shrinkage has to be considered for simulating the creep tests.
The material parameters referring to shrinkage and creep are calibrated on the basis of test series 4 including only creep test series 4/2. Subsequently, the performance of the shotcrete models is evaluated by simulating creep test series 4/1 and 3 employing the identified parameters.
Parameter identification from the shrinkage test on the unsealed specimen in test series 4 by the method of least squares yields the ultimate shrinkage strain = -0.0019 (with = 1.0) and the shrinkage half time = 32 for the Meschke model and the SCDP model. The respective values for the Schädlich model are obtained as = -0.0015 and = 8.3 . Although as well as and have the identical physical meaning for all models, different values are estimated. This is the consequence of the different temporal evolution laws of the shrinkage models, which together with the short time span of the experimental shrinkage data, result in the identification of different parameters, depending on the employed shrinkage law.
The environmental conditions, required for the Gawin model, are assumed as and . The same heat and mass transfer coefficients, porosity, hygral and thermal parameters as described in the previous subsection for the experiments by Huber are applied. Since drying induces creep in the Gawin model, the drying behavior cannot be calibrated independently from the creep parameters, which are presented for the subsequent numerical simulation of the creep tests.
a shows the results of the calibration of the shotcrete models based on test series 4 in [17
]. Figure 4
b contains the validation of the shotcrete models based on test series 3 in [17
Creep tests on unsealed specimens were conducted simultaneously with the shrinkage tests. The applied load was increased multiple times during the tests lasting approximately 28 days in total. Periods of constant stress levels lasted only for a few hours or days before the load was increased further. The start time of initial loading and the loading sequence was different for each test series. The individual loading sequences are listed in Table 3
. Test series 3 includes identical creep tests on three specimens, here denoted as specimens a, b and c. In test series 4, the two different loading sequences 4/1 and 4/2 were applied to two specimens each. The four specimens of test series 4/1 and 4/2 are denoted as a and b, and c and d, respectively. A third specimen tested in test series 4/1 is not considered here, as the excessive measured total strain indicates a faulty specimen.
The creep parameters of the three models are exclusively identified from test series 4/2, and are subsequently used to simulate test series 3 and 4/1. The Young’s modulus at the age of one day,
, is specified in [17
for test series 4. The Young’s modulus at the age of 28 days,
, is computed as the mean value of the two values presented in [17
= 11,580 MPa. The uniaxial compressive strength at the age of one day is reported as
and the compressive strength at the age of 28 days was measured in tests series 4 on three specimens, from which the mean value is computed as
The ratio of the yield stress over the compressive strength
, determined from uniaxial compression tests, strongly influences the creep behavior of the Meschke model due to the viscoplastic formulation. Since identification from the creep tests leads to a ratio
tending to zero,
is estimated as
for all models, which is the lower bound of the range proposed in [23
]. The plastic strains at uniaxial compressive peak stress for different shotcrete ages,
, are chosen as the default values proposed in [4
Regarding the creep half-time parameter
of the Schädlich model, identification simultaneously with the creep coefficient
leads to an ill-posed problem due to the short time span of the available experimental data. Hence,
is chosen as
, supported by the evaluation of different values for
employed for the present parameter identification scheme. For comparison,
is proposed in [42
]. The viscoelastic compliance parameters
of the SCDP model are computed according to the guidelines [25
], which results in
follows from the measured value of
, with the effective Young’s modulus computed for a time period of
. Further creep parameters are identified from both specimens of test series 4/2 considering only the time-dependent strain during the first level of the sustained stress, yielding the viscosity parameter η
for the Meschke model, the creep coefficient
= 1.21 for the Schädlich model, and the flow compliance parameter
for the SCDP model. For the Gawin model, the estimated parameters governing hydration, the evolution of uniaxial compressive strength and effective Young’s modulus are
. Using this parameter set, the effective Young’s modulus evaluated for a time period of
is in the range of 7600 MPa to 7780MPa at the age of 1 day and 11,440 MPa to 11,700 MPa at the age of 28 days. The corresponding values for the uniaxial compressive strength are in the range of 8.57 MPa to 8.9 MPa at the age of 1 day and 16.41 MPa to 17.06MPa at the age of 28 days. Maximum values are always obtained at the center of the sample, while minimum values are located at the corners where drying and cooling slows down the hydration process. Parameters governing the viscous creep are
. The parameters in the effective stress relationship (27) are estimated as
. They are obtained based on the drying shrinkage test. In the range above 50% water saturation, which is present in the computation, these values model a response which is close to the response of ordinary concrete shown in [10
shows the results of the calibration of the shotcrete models, based on the experimental data of the first sustained stress level of test series 4/2. Due to the viscoplastic formulation of the Meschke model, the viscosity parameter η
only controls the rate of the creep strain in the hardening regime, but not its magnitude. This explains the discrepancies between measured and predicted total strain for the Meschke model. The validation of the shotcrete models based on the further load levels of test series 4/2 is shown in Figure 6
. Figure 7
and Figure 8
contain the validation of the shotcrete models based on the experimental data of test series 4/1 and 3, respectively.
It can be seen that the numerical results by the SCDP model and the Gawin model agree very well with the experimental data, while the Schädlich model and the Meschke model underestimate the measured total strain. In the Gawin model, the stress-dependent amplification factor for the creep strain at high stress levels, which causes the fast strain increase in the highest load steps of series 4, is very sensitive to the value of uniaxial compressive strength.