1 Introduction

Though creep of concrete has been a subject of interest for several decades, attention has mainly been devoted to creep under a uniaxial stress condition. However, in reinforced and prestressed structures, a three-dimensional state of stress generally exists, which can complicate the response of the structure. To understand the behavior of concrete under multiaxial compression, the Poisson’s ratio of the viscoelastic material plays a crucial role to determine the long-term deformation response and durability performance of concrete [1].

Some investigators have studied the viscoelastic/viscoplastic Poisson’s ratio (VPR) of concrete, but the results reported from different studies are contradictory. For example, it has been suggested that VPR is an increasing, decreasing, and constant function of time in separate studies. Such uncertainty arises mainly from the fact that the VPR is measured or calculated differently by different researchers. Ross was the first to conduct creep tests on concrete under 2D loading and suggested that the creep Poisson’s ratio (CPR)Footnote 1 is close to zero [2]. His experiments showed that creep in the direction of major stress in 2D testing reaches a magnitude of the same order as that under simple 1D stress of same intensity. A few years later, Gopalakrishnan reported that under multiaxial stress conditions, the CPR was lower than the uniaxial Poisson’s ratio and that there was no variation in CPR with time [3]. The CPR was calculated separately for each direction using the knowledge of the uniaxial compliance. Gopalakrishnan also argued that the CPR was a function of the magnitude of stress.

Jordaan and Illston calculated CPR as the ratio of the total mechanical strains (i.e., the sum of elastic and creep strains) in the axial and lateral directions [4]. They proposed four different expressions for calculating CPR under different loading conditions (uniaxial, biaxial, triaxial with uniaxial system, triaxial with octahedral shear stresses). The effective Poisson’s ratio for all cases remained constant and equal to the Poisson’s ratio in an elastic state [4]. In 1974, Parrott measured the lateral strains from uniaxial tests on cement paste to determine the Poisson’s ratio [5]. He used the creep strains (the total strain minus the elastic and shrinkage strains) to calculate the CPR. The CPR was found to be a constant value equal to 0.13 [5]. Kesler found that the CPR of concrete that was sealed during loading to be almost equal to the elastic Poisson’s ratio, however it was considerably smaller if allowed to dry under load [6]. Lakes demonstrated that composite structures may exhibit increasing or decreasing Poisson’s ratio with time [7]. He also proposed that time dependent VPR need not be monotonic in nature, where a composite can be constructed that can have a decreasing Poisson’s ratio with time initially followed by an increasing function. Hilton cited five different expressions for time-dependent Poisson's ratio and identified VPR’s strong dependence on stress histories [8]. Grasley and Lange computed the VPR of sealed cement paste using the correspondence principle. They found that under multiaxial loading, the VPR of cement paste is relatively constant with time and then gradually increases as dilatational compliance comes to a halt [9].

More recently in 2015, Aili used various multiaxial creep test data from literature to show the difference between the CPR and relaxation Poisson’s ratio (RPR – equivalent to the VPR) [10]. In spite of the two Poisson’s ratio not being equal [11, 12] the initial and long-term asymptotic values and their corresponding time derivatives were found to be the same [10]. In another study, Aili found the VPR for a mature concrete to be constant and ranging between 0.15 and 0.20 [13]. Charpin conducted a 10-year concrete creep study under uniaxial and biaxial conditions in which the evolution of the CPR was discussed. Their experimental data showed that the assumption of a constant Poisson ratio for concrete is reasonable [14]. In 2017, Charpin and Sanahuja established through examples, both theoretical and practical, that any evolution of Poisson’s ratio: increasing, decreasing or non-monotonic is possible for concrete [15]. In summary, there is a large scatter in the reported VPR from different studies at room temperatures. A possible reason that could partly explain this large scatter in data is that the experiments were performed under varying test conditions. Also, the laboratory measurement of axial and lateral strains is challenging as they highly depend on the resolution of strain gages used. Overall, these conditions make multiaxial creep tests strenuous and time-consuming, thereby serving as a motivation for researchers to develop more effective methods to evaluate VPR.

Data on the Poisson’s ratio of concrete at elevated temperatures are scarce and limited. At ambient temperature, the Poisson’s ratio of concrete can vary between 0.15 and 0.20 [13, 16]. A study by Ehm in 1985 [17], suggests that the Poisson’s ratio decreases with increasing temperatures due to weakening of the microstructure by breakage of bonds at higher temperatures [17]. For a concrete under confining pressure, as it would be in many nuclear power plant concrete structures, it has been hypothesized that Poisson’s ratio at elevated temperature would be about the same as at room temperature [18].

All the above studies are either for elastic Poisson’s ratio at elevated temperatures or VPR (or CPR) at room temperature. The authors have not found any data in the literature on VPR for concrete, mortar, or cement paste at elevated temperatures. In the current study, the authors utilize a confined creep test that can fully capture the 3D constitutive properties of cement mortar in a single experiment. As concrete creep is entirely due to the creep within the cement paste phase, since the behavior of aggregates is typically linear elastic [19], the sensitivity of concrete creep to both temperature and multi-axial loading can be captured by cement mortar experiments. The focus on cement mortar samples rather than concrete enables the use of smaller sized specimens with enhanced measurement resolutions. For this purpose, a smaller creep frame is designed that can fit in climatic chambers to facilitate testing at elevated temperature. The confined creep test allows the simultaneous measurement of bulk and shear compliance, which is used to determine the uniaxial compliance and VPR through intermodulus conversion via the correspondence principle [9]. The temperature-dependency of VPR is analyzed by running confined creep tests at room temperature (i.e., 20 °C) and elevated temperature (i.e., 60 °C). Hence, this research aims to improve the understanding of the viscoelastic behavior of cement mortar under multiaxial loading as a function of time and temperature.

2 Experimental design

2.1 Mix design

The cement mortar mix design selected for this study closely resembles the Électricité de France (EDF) VeRCoRs mortar mix. The mortar samples were prepared using Type I/II cement and river sand. The river sand was sieved to pass the 2.38 mm sieve and dried for 24 hours before mixing. The water to cement ratio (w/c) by mass for the mix was 0.52 in Saturated Surface Dry (SSD) condition and the sand to cement ratio by mass was 2.12. A water-reducing admixture ‘Pozzolith 80’ was administered at a dosage of 4.22 mL per kilogram of cementitious materials. The mixture proportions used are shown in Table 1 and referenced in EDF (EDF [20]).

Table 1 Mixture proportions (SSD condition)

2.2 Creep frame design

Similar to the standard ASTM C512 concrete creep frames, a unique miniature version of the creep frame was designed and fabricated exclusively for cement mortar samples of dimensions 50 mm radius × 100 mm height (2 in. radius × 4 in. height) as shown in Fig. 1. The total height of the scaled down frame is only 45 cm (18 in.) as opposed to approximately 180 cm (6 ft.) tall standard concrete creep frames. The diameter of the frame is 10 cm (4 in.), making the frames easier to be placed inside temperature ovens and climate chambers. The creep frame has a compression spring at the bottom of the frame that helps to maintain a nearly constant load. Above the spring is a plate with a ball bearing at the center to ensure minimum eccentricity in loading. An inline load cell is placed just below the sample to record the load levels in the frame. Although load levels are intended to be constant during a creep test, the actual load level was recorded periodically to account for any load loss. A 5000 kg mini hydraulic jack was used to apply the initial axial force. Threaded rods and nuts were used in the frame to maintain compression of the spring and a relatively constant load after removal of the hydraulic jack.

Fig. 1
figure 1

Miniaturized compressive creep test frame for cement mortar. This frame is 45 cm (18 in.) in total height and is a scaled down version of ASTM standard concrete creep frame which is approximately 180 cm (6 ft.) in height

Cement mortar was mixed in accordance to ASTM C305-99 and immediately cast into a 304L stainless steel confining tube of 2 mm (0.07 in.) thickness (54 mm outer diameter and 50 mm inner diameter) and 100 mm (4 in.) height. Embedded vibrating wire gages with gage length of 50 mm (2 in.) from Geokon were mounted axially at the center of the confining tubes using fishing line before pouring the mix. The cylinders were filled in three equal increments and tapped after each increment to minimize air voids. Both ends of the confining tube were sealed using tightly fitted caps. After 24 hours, the end caps were removed, and the sample was pushed slightly outside the tube on either end separately to smoothly cut the surface using a diamond blade wet saw. This procedure ensures that the bond between the sample and the confining tube is broken and the contact between them can be approximated as frictionless. Thus, the radial displacements between the inner surface of the steel tube are consistent with the radial displacements of the outer surface of the mortar.

To measure the hoop strain, 4 sets of foil strain gages (C2A-13-250LW-120) from Micro Measurements were mounted on the outer radial surface of the steel tube. The foil strain gages were placed diametrically opposite to each other and in the circumferential direction. A steel loading block with a diameter of 49 mm (1.9 in.) and height of 50 mm (1.9 in.) was placed on both ends of the sample but did not contact the confining tube. The loading block diameter was slightly less than the inner diameter of the confining steel tube so that it can easily slide through the tube while also ensuring uniform compressive stress throughout the sample cross-section.

The cement mortar specimens were cured at room temperature for 28 days before starting the test. In addition, the elastic properties of stainless steel are the same at 60 °C as at 20 °C [21]. At 28 days, the samples were loaded using a hydraulic jack to a constant load of approximately 1905 kg (4200 lbs), which corresponded to 25% of 28-day compressive strength of mortar, f’c = 37.23 MPa (5400 psi) at 28 days. At this loading age and stress magnitude, the mature cement mortar was assumed to be a non-aging, linear viscoelastic material. The load was approximated as a stepwise load function given the very short time span of load application relative to the overall duration of the creep test. The axial strain from the vibrating wire gage as well as the load readings from the load cell were recorded every 30 minutes using a CR300 Data logger, AM16/32B Multiplexer and a 2-Channel Vibrating-Wire Analyzer (AVW200) from Campbell Scientific. The foil strain gages that read hoop strain were connected to a Student D4 data acquisition system from Micro Measurements using quarter bridge circuits. As mentioned earlier, creep tests were run at 20 °C and 60 °C. At 60 °C, the creep frame and the sample were heated to the test temperature until the thermal strains in the sample stabilized (indicating uniform temperature in the sample), which took approximately 30 minutes to 1 hour. Since, the cement mortar samples were cured at room temperature for 28 days, before heating the samples to test temperature, the compressive strength of the 60 °C samples was presumed unchanged from the samples tested at 20 °C [22]. The experiments were conducted in environmental chambers maintaining a constant temperature with  ± 1 °C accuracy Fig. 2. The relative humidity was consistent at 50% in the 20 °C chamber and below 10% in the 60 °C chamber.

Fig. 2
figure 2

Confined creep test setup using the miniaturized creep frame placed inside an environmental chamber maintaining constant temperature and humidity

3 Computation of viscoelastic material properties

The confined creep apparatus was first utilized by Ma and Ravi-Chandar and subsequently by Park and Roy [23, 24] to obtain the bulk and shear linear viscoelastic compliance functions simultaneously on the same specimen, under constant environmental conditions. Grasley and Lange [9] were the first to use such an experimental setup on cementitious materials when they studied cement paste creep at room temperature. In this study, the confining steel cylinder with an inner radius \(a\) of 25 mm (1 in.), an outer radius \(b\) of 27 mm (1.06 in.) and a height \(h\) of 100 mm (4 in.) was used. The material properties of the confining steel cylinder are known. The Young’s modulus of stainless steel \((E^{c} )\) is 193 GPa and Poisson’s ratio \((\nu^{c} )\) is 0.29. The elastic properties of stainless steel are the same at 60 °C as at 20 °C [21]. The confined test set-up in the cylindrical polar coordinate system is shown in Fig. 3.

Fig. 3
figure 3

Confined compressive creep test set-up with inner radius \(a\) of 25 mm (1 in.), an outer radius \(b\) of 27 mm (1.06 in.) and a height \(h\) of 100 mm (4 in.)

The axial stress, \(\sigma_{zz}\), experienced by the cement mortar sample under a constant load was calculated by dividing the load applied by the cross-sectional area of the sample such that

$$\sigma_{zz} = \frac{{{\text{Load}}\;{\text{applied}}}}{{\pi a^{2} }}.$$

The total axial strain as a function of time, \(\upvarepsilon _{a} (t)\), was recorded using a vibrating wire gage. The mechanical axial strain, \({\upvarepsilon }_{zz} (t)\), was calculated as

$$\varepsilon_{zz} (t) = \varepsilon_{a} (t) - \varepsilon^{{\text{sh}}} (t),$$

where \({\upvarepsilon }^{{\text{sh}}} (t)\) is the free shrinkage strain. Since the specimens were sealed by the confining cylinder, the free shrinkage strain was greatly reduced. Fig. 4 shows the free strain measured in a sample at room temperature confined by a stainless steel tube to be far lower than that of a similar sample wrapped with a layer of adhesive backed aluminum foil. Nevertheless, the free strain was subtracted from the total axial strain as shown in Eq. (2).

Fig. 4
figure 4

Free strain comparison of a sample in a stainless steel tube to sample wrapped with Al foil at 20 °C. Here \(t\) refers to the present age and \(t^{\prime}\) refers to the age when confined creep test started (i.e., 28 days)

When an axial stress is applied, the specimen tends to expand in the transverse direction due to Poisson’s effect, resulting in a positive (or tensile) radial displacement \((u_{r} )\). However, the confining cylinder restrains this deformation with a negative (or compressive) radial stress \((\sigma_{{\text{rr}}} )\).

From the continuity conditions at the interface

$$\sigma_{{\text{rr}}} (a) = \user2{ }\sigma_{{\text{rr}}}^{c} (a)$$


$$u_{r} (a) = u_{{\text{rr}}}^{c} (a),$$

where superscript c denotes confining cylinder.

By measuring the hoop strain, \((\upvarepsilon _{h} )\), on the outer surface of the confining cylinder using the foil strain gages, the radial displacement and radial stress of the confining cylinder on the inner surface, \(u_{r}^{c}\) and \(\sigma_{rr}^{c}\), respectively, were computed by the Lamé solution as

$$u_{r}^{c} = \frac{{\varepsilon_{h} }}{2}\left[ {\left( {1 - \nu^{c} } \right)a + \left( {1 - \nu^{c} } \right)\frac{{b^{2} }}{a}} \right]$$


$$\sigma_{rr}^{c} (t) = - \frac{{b^{2} - a^{2} }}{{2a^{2} }}E^{c} \varepsilon_{h} (t).$$

The stress induced radial strain, \({\upvarepsilon }_{{\text{rr}}} (t)\), was calculated by dividing the radial displacement, \(u_{r}^{c}\), by inner radius, \(a\), and subtracting the free shrinkage strain such that

$$\varepsilon_{{\text{rr}}} (t) = \frac{{\varepsilon_{h} (t)}}{2}\left[ {\left( {1 - \nu^{c} } \right) + \left( {1 - \nu^{c} } \right)\frac{{b^{2} }}{{a^{2} }}} \right] - \varepsilon^{{\text{sh}}} (t).$$

The axisymmetric strain-displacement relation gives \(\varepsilon_{{\text{rr}}} = \varepsilon_{{\uptheta \uptheta }}\) and equilibrium equations require that radial and hoop components of stress are equal, i.e., \(\sigma_{rr} = \sigma_{\theta \theta }\), where \(r\) and \(\theta\) represent radial and tangential components, respectively. Since eqs. (1), (2), (6) and (7) representing the axial stress, axial strain, radial stress and radial strain, respectively, form the principal components of stress and strain, the full 3D constitutive response of the material was obtained from the confined experiment.

The deformation of the specimen was then separated into dilatational and deviatoric components. The volumetric stress, \(\sigma_{m}\), in the specimen was determined as one-third the sum of the three normal stress components such that

$$\begin{aligned} {{\varvec{\upsigma}}}_{{{\mathbf{m}} }} \left( {\mathbf{t}} \right) = & \frac{{{{\varvec{\upsigma}}}_{{\text{kk} }} \left( {\mathbf{t}} \right)}}{3} \\ = & \frac{1}{3}\left[ {{{\varvec{\upsigma}}}_{{\text{zz} }} \left( {\mathbf{t}} \right) + {{\varvec{\upsigma}}}_{{\text{rr} }} \left( {\mathbf{t}} \right) + {{\varvec{\upsigma}}}_{{\uptheta \uptheta }} \left( {\mathbf{t}} \right)} \right] \\ \user2{ } = & \frac{1}{3}\left[ {{{\varvec{\upsigma}}}_{{\text{zz} }} \left( {\mathbf{t}} \right) + 2{{\varvec{\upsigma}}}_{{\text{rr} }} \left( {\mathbf{t}} \right)} \right]. \\ \end{aligned}$$

Since the deformation gradients are small, the volumetric strain, \(\varepsilon_{kk}\), was approximated as the sum of three normal strains such that

$$\begin{aligned} \varepsilon_{{\text{kk}}} (t) = & \left[ {\varepsilon_{{\text{zz}}} (t) + \varepsilon_{{\text{rr}}} (t) + \varepsilon_{{\uptheta \uptheta }} (t)} \right] \\ = & \left[ {\varepsilon_{{\text{zz}}} (t) + 2\varepsilon_{{\text{rr}}} (t)} \right]. \\ \end{aligned}$$

The deviatoric stress, \(\tau_{e}\), and deviatoric strain, \(\gamma_{e} \user2{ }\), from the confined compression test were computed using

$$\tau_{e} (t) = \frac{1}{\sqrt 3 }\left| {\sigma_{{\text{zz}}} (t) - } \right.\left. {\sigma_{{\text{rr}}} (t)} \right|$$


$$\gamma_{e} (t) = \frac{2}{\sqrt 3 }\left| {\varepsilon_{{\text{zz}}} (t) - \left. {\varepsilon_{{\text{rr}}} (t)} \right|} \right..$$

The constitutive equations for a non-aging, linear viscoelastic material, expressed in terms of dilatational and deviatoric components, is given by

$$\varepsilon_{{\text{kk}}} (t) = \mathop \smallint \nolimits_{0}^{{\mathbf{t}}} B\left( {t - t^{\prime}} \right)\frac{{\partial \sigma_{m} (t^{\prime})}}{\partial t^{\prime}}dt^{\prime}$$


$$\gamma_{e} (t) = \mathop \smallint \nolimits_{0}^{{\mathbf{t}}} L(t - t^{\prime})\frac{{\partial \tau_{e} (t^{\prime})}}{\partial t^{\prime}}dt^{\prime},$$

where \(B(t)\) and \(L(t)\) are the bulk and shear compliances of the material, respectively. Here \(t\) refers to the present time and \(t^{\prime}\) is the dummy integration time variable. Once the bulk and shear compliances were found, the uniaxial creep compliance, \(J(t)\), was computed using intermoduli conversion via the correspondence principle such that

$$\overline{E} (s) = \frac{{9\overline{K} \user2{ }(s)\overline{G} (s)}}{{3\overline{K} \user2{ }(s) + \overline{G} (s)}}$$


$$\overline{J} \left( {\mathbf{s}} \right). \overline{E} \left( {\mathbf{s}} \right) = \user2{ }\frac{1}{{s^{2} }},$$

where \(\overline{E} (s)\), \(\overline{K} (s)\) and \(\overline{G} (s)\) are Laplace transformed relaxation Young’s, bulk, and shear modulus functions, respectively, and \(s\) is the complex variable in the Laplace domain. The Laplace transformed uniaxial creep compliance and the corresponding Laplace transformed relaxation moduli are each related as shown in eq. (15).

4 Results and discussion

4.1 Initial loading

The confined creep test was performed on a specimen each at 20 °C and 60 °C. The axial stress/strength ratio applied was 0.25, which lies within the linearity range of the material according to the study by Neville and Dilger, who demonstrate that non-linearity of cement mortar arises only beyond a stress/strength ratio of 0.80 [25]. Figure 5 shows that when the compressive axial load was applied on the sample, the compressive axial strain was accompanied by an increase in tensile hoop strain.

Fig. 5
figure 5

Variation of axial strain and hoop strain at initial loading. Data from specimen at 20 °C.

As outlined earlier, using the measured hoop strain and material properties of the confining cylinder, the principal components of stress and strain were determined. Fig. 6 shows the volumetric (or bulk) and shear stresses plotted against their corresponding volumetric (or bulk) and shear strains at 20 °C and 60 °C.

Fig. 6
figure 6

Volumetric (or bulk) and shear stress-strain data at initial loading at 20 °C and 60 °C.

It is clear from Fig. 6 that the samples exhibited linear behavior at the time of initial loading. The slope of these curves represents the elastic bulk and shear modulus of cement mortar at the respective temperatures. The elastic bulk modulus, K, was calculated to be 13.6 GPa and the elastic shear modulus, G, was calculated to be 11.6 GPa at 20 °C. Using the intermodulus conversion, the elastic Young’s modulus, E, was calculated to be 27.0 GPa. To validate the calculations, the E of three mortar cylinders (100 mm × 200 mm) were measured in accordance with ASTM C469. The average measured Young’s modulus of the samples at 28 days was 25.4 ± 1.1 GPa. The predicted and measured elastic modulus values were hence found to be in reasonable agreement. Similarly, at 60 °C from Fig. 6, K was calculated to be 11.0 GPa and G was calculated to be 9.4 GPa resulting in E of 21.8 GPa using intermodulus conversion. It is unclear why the Young’s modulus was lower at 60 °C than at 20 °C, though other researchers have measured reductions in stiffness of concrete [26] and cement paste [27] at elevated temperatures around 50 °C.

4.2 Analysis of methods for determining \(B(t)\) and \(L(t)\) from confined creep test

The deformation from the confined experiment was separated into dilatational and deviatoric parts for the determination of dilatational and deviatoric compliance functions. Although the axial load was maintained relatively constant throughout the experiment, the volumetric and shear stress varied as a function of time and temperature as shown in Fig. 7.

Fig. 7
figure 7

The volumetric and shear stress as a function of time during the confined creep experiment at 20 °C and 60 °C. Here \(t\) refers to the present age and \(t^{\prime}\) refers to the age when confined creep test started (i.e., 28 days)

It is clear from Fig. 7 that shear stress showed significant decay with time compared to the volumetric stress at both temperatures. There was a 37% decay in shear stress magnitude compared to 24% decay in volumetric stress magnitude after 90 days of testing at room temperature. On the other hand, at 60 °C, there was a 49% decay in shear stress compared to 28% decay in volumetric stress after 90 days.

Since volumetric and shear strains are evolving along with the corresponding stresses during the experiment, a more informative way to assess the confined creep test results is to determine the bulk and shear compliance functions, since these functions are ostensibly independent of either stress or strain history. To understand the significance of stress decay in the confined test, the compliance function was estimated using three methods. The first method (simple division method) is an approximate method whereby load history effects are neglected, and strains are simply divided by their respective stress values such that

$$B(t) \approx \frac{{\varepsilon _{{kk}} (t)}}{{\sigma _{m} (t)}}{\mkern 1mu}\; {\text{and}}\;{\mkern 1mu} L(t) \approx \frac{{\gamma _{e} (t)}}{{\tau _{e} (t)}}$$

The second method (constant stress method) completely neglects the time variance in the stresses and considers the stress to be constant and equal to the initially applied stresses such that

$$B(t) \approx \frac{{\varepsilon_{kk} (t)}}{{\sigma_{m0} }}\;{\text{and}}\;L(t) \approx \frac{{\gamma_{e} (t)}}{{\tau_{e0} }}.$$

The third method (“fitted”) employs the constitutive equations for a linear viscoelastic material to derive the compliance functions accounting for the time-dependency of the stresses and the history dependence of the strains. In order to determine \(B(t)\) and \(L(t)\) using the constitutive expression given in eqs. (12) and (13), it is necessary to fit the measured stress history to a time dependent function, take the derivative of that function (in terms of the dummy time variable), multiply the derivative by a presumed function for \(B(t)\) and \(L(t)\) – including phenomenological fit coefficients – and then integrate the product over time. The resulting time dependent function was fit to the measured strain data to determine the phenomenological fit coefficients included in \(B(t)\) and \(L(t)\). A logarithmic function was used to represent the compliance functions with retardation time set as 1 day, 10 days, 100 days and so on. This is the most fundamentally accurate of the three methods to determine the compliance functions. The other methods may be used for estimating compliance but do not yield precise calculations.

Figure 8 shows the graph of the compliance function obtained using these three methods. The \(B(t)\) and \(L(t)\) clearly increased at higher temperature due to larger creep strains. There was insignificant variation in the \(B(t)\) and \(L(t)\) calculated using the three methods at 20 °C, implying that either of the two approximate methods may be used to simplify the calculations when the experiments are performed at room temperature. At higher temperatures, however, the magnitude of creep in the samples caused force relaxation in the loading frame leading to greater time dependency of the stresses. Assuming the stress to be constant in such cases while calculating \(B(t)\) and \(L(t)\) can lead to significant errors. There was a 30% and 47% error while using the constant stress method to estimate \(B(t)\) and \(L(t)\), respectively, at 60 °C after 90 days of loading.

Fig. 8
figure 8

The bulk and shear compliance functions calculated using the three methods, a simple division, b assuming load to be constant, c, fitting the compliance in the constitutive equation (most accurate)

Once the \(B(t)\) and \(L(t)\) were computed, they were transformed in to the Laplace domain to find the Laplace transformed bulk modulus, \(\overline{K} \left( {\mathbf{s}} \right)\) and shear modulus, \(\overline{G} \left( {\mathbf{s}} \right)\) from the relations

$$\bar{B}\left( s \right).~\bar{K}\left( s \right) = ~\frac{1}{{s^{2} }}\;{\text{and}}\;\bar{L}\left( s \right).~\bar{G}\left( s \right) = ~\frac{1}{{s^{2} }}.$$

The Laplace transformed relaxation modulus, \(\user2{ }\overline{E} (s)\), and subsequently the Laplace transformed uniaxial creep compliance, \(\overline{J} (s)\), were found using the intermodulus conversion expression shown in eqs. (14) and (15). The inverse Laplace transform was then used to determine the compliance in the time domain. A graph depicting the predicted compliance through the intermodulus conversion compared to the measured compliance from uniaxial creep data at both temperatures are shown in Fig. 9. It can be seen that the predicted compliance was reasonably close to the measured compliance data at 20 °C, whereas at higher temperature, the converted uniaxial compliance was significantly higher than the directly measured uniaxial compliance. A possible explanation is that the viscoelastic response is nonlinearly dependent on strain energy magnitude,this would be in keeping with the findings of Li et al., who simulated relaxation of cement pastes due to dissolution of hydration products [28]. In that work, dissolution was modeled as a function of local strain energy magnitude, which implies that application of confining stress can serve to enhance the axial relaxation rate. Experimental studies have verified that dissolution of minerals is accelerated under high local stress [29, 30, 3132]

Fig. 9
figure 9

The predicted uniaxial creep compliance using intermodulus conversion.

For the tests evaluated in Fig. 9 conducted at 20 °C, the strain energies were 0.454 kPa and 1.381 kPa for the uniaxial and confined experiments, respectively. The strain energies for the tests evaluated at 60 °C were 0.625 kPa and 6.764 kPa for the uniaxial and confined experiments, respectively. The total strain energy in the system is significantly higher in the case of multiaxial loading than uniaxial loading condition for the same stress/strength ratios. Hence the rule of thumb of linearity with respect to stress level below 30%-40% of the strength does not necessarily hold true for multiaxial loading condition. With higher strain energy in multiaxial loading, non-linearity could be a reason for difference in measured versus predicted uniaxial creep compliance using intermodulus conversion.

4.3 Viscoelastic Poisson’s ratio

As alluded to previously, there is contradiction in the literature regarding the measurement and prediction of VPR of concrete with time. This uncertainty arises mainly from the way researchers define the VPR function.

As shown by Kassem et al. [33], in a displacement controlled experiment such as a stress relaxation test where the input axial strain, \(\varepsilon_{11}\), is a step function represented as \(\user2{ }\varepsilon_{11} (t) = \varepsilon_{0} H(t)\), where \(\varepsilon_{0}\) is a constant and \(H(t)\) is the Heaviside function, the output transverse strain \(\varepsilon_{22} (t)\) is given by

$${\varvec{\varepsilon}}_{{22\user2{ }}} (t) = \user2{ } - {{\varvec{\upnu}}}\left( {\mathbf{t}} \right) {{\varvec{\upvarepsilon}}}_{0} .$$

VPR can then be calculated as

$${{\varvec{\upnu}}}\left( {\mathbf{t}} \right) = - \frac{{{\varvec{\varepsilon}}_{{22\user2{ }}} (t)}}{{{\varvec{\varepsilon}}_{0} }}.$$

On the other hand, in a load controlled experiment, such as a creep test where the input axial stress \(\sigma_{11}\) is a step function represented as \(\sigma_{11} (t) = \sigma_{o} H(t)\), where \(\sigma_{0}\) is a constant, the output axial strain is given by

$${\varvec{\varepsilon}}_{11} (t) = D(t){{\varvec{\upsigma}}}_{0} .$$

The corresponding output transverse strain is given by

$${\varvec{\varepsilon}}_{22} (t) = - \mathop \int \nolimits_{0}^{{\mathbf{t}}} {{\varvec{\upnu}}}\left( {t - t^{\prime}} \right)\frac{{\partial {{\varvec{\upvarepsilon}}}_{11} (t^{\prime})}}{\partial t^{\prime}}dt^{\prime}.$$

In order to determine \(\nu (t)\) using Eq. (22), it is necessary to fit the measured axial strain history to a time dependent function, take the derivative of that function (in terms of the dummy time variable), multiply the derivative by a presumed function for \(\nu (t)\) —including phenomenological fit coefficients—and then integrate the product over time. The resulting time dependent function is then fit to the measured transverse strain data to determine the phenomenological fit coefficients included in \(\nu (t)\). Many researchers have determined a “creep Poisson’s ratio” for concrete by neglecting the history dependence denoted by the convolution integral in Eq. (22),such a creep Poisson’s ratio is a function of the stress or strain history of the material and is thus not a constitutive property like the VPR determined in Eq. (20) or (22).

In case of a specimen subjected to a 3D state of stress, such as a confined creep test, the VPR may be determined using the intermoduli conversion in the Laplace domain according to

$$s.\user2{ }\overline{\nu } (s) = \user2{ }\frac{{3\overline{K} (s) - 2\overline{G} (s)}}{{6\overline{K} (s) + 2\user2{ }\overline{G} (s)}}.$$

In this study, using the Laplace transformed bulk and shear modulus, the Laplace transformed VPR was determined from Eq. (23). Then, the Laplace transformed VPR was inverted to the time domain to obtain the VPR of the viscoelastic material (see Fig. 10. Mathematica was used to perform the analyses.

Fig. 10
figure 10

Variation of viscoelastic Poisson’s ratio of cement mortar as a function of time

The viscoelastic Poisson’s ratio of cement mortar was found to be a relatively constant value of 0.16 at 20 °C and there was a slight increase from 0.17 to 0.19 over a period of 100 days after initial loading at 60 °C. It is interesting to note that the elastic Poisson’s ratio is slightly higher at elevated temperatures since the shear modulus decreased more significantly than the bulk modulus when the temperature was increased to 60 °C compared to 20 °C.

If both the dilatational and deviatoric compliance functions show a similar trend (i.e., same rate at all times), then VPR can be considered a constant function with time. From Fig. 10, it can be seen that the VPR is largely not time-dependent at 20 °C but at 60 °C, where the rate of increase of deviatoric compliance was higher than the rate of increase of dilatational compliance, VPR was found to increase slightly as a function of time. The same phenomenon is discussed by Bernard et al. and Grasley and Lange, where the reduction in porosity in the sample was suggested to decrease the rate of change of dilatational compliance with time versus the rate of change of deviatoric compliance [9]. If strain energy magnitude strongly affects creep or relaxation rates in concrete as hypothesized in the previous section, it is important to note that – since the VPR is still relatively constant at 60 °C, where the strain energy was significant – the effect of strain energy magnitude on both deviatoric and dilatational relaxation or creep rates is almost the same. The implication is that, while relaxation rates might depend on strain energy magnitude, the VPR can be approximated as largely independent of strain energy magnitude, at least for the magnitudes considered in this study.

To analyze uncertainty in the determination of the Poisson’s ratio, the authors used Monte Carlo approach to simulate a range of possible \(\nu (t)\) values. For that purpose, the spring constants in K(t) and G(t) were expressed as a random variate of a normal distribution function. The coefficient of variation for the spring constants was obtained from the uniaxial creep test [34], which had three replicates for each test temperature. Here, the authors have presumed that the coefficient of variation is consistent for E(t), K(t), and G(t). Using random variate for spring constants, a dataset of Poisson's ratio was obtained at different time steps. The Poisson's ratio at both 20 °C and 60 °C showed little variation within one standard deviation (The Poisson’s ratio at 20 °C was a value of 0.16±0.012 at t=0 and 0.15±0.003 at t=100. Similarly, at 60 °C, the Poisson’s ratio was a value of 0.17±0.018 at t=0 and 0.19±0.003 at t=100).

5 Conclusions

The study describes a simple and cost-effective experiment to conduct a confined creep test on cement mortar using a miniaturized creep frame. This creep test was used to quantify the 3D creep response through direct determination of the full stress and infinitesimal strain tensors. Consequently, the viscoelastic compliance functions and Poisson’s ratio of cement mortars were investigated at two different temperatures. The deformation from the confined experiment was separated into dilatational and deviatoric parts for the determination of dilatational and deviatoric compliance functions. It was observed that the dilatational and deviatoric compliance functions showed a similar trend at 20 °C whereas, at 60 °C, the rate of change of deviatoric compliance was slightly higher compared to dilatational compliance. Using the intermodulus conversion, the viscoelastic Poisson’s ratio of cement mortar was found to be a nearly constant value of 0.16 at 20 °C and increased from 0.17 to 0.19 over a period of 100 days after initial loading at 60 °C.

Another critical finding was that the uniaxial compliance calculated from the bulk and shear relaxation moduli was higher than measured directly, with the magnitude of the difference seeming to depend on the magnitude of the strain energy present in the test. Though more testing is recommended, it is hypothesized that the higher strain energy increases the rate of dissolution of hydration products and thus the creep resulting from the requisite stress redistribution. The practical implication of the dependence of the compliance function on strain energy is that the application of moderate lateral stresses can induce non-linearity of the viscoelastic constitutive behavior; this means that one might not be able to utilize creep compliances determined experimentally in a uniaxial test to predict the response in structures with 2D or 3D states of stress. Interestingly, the results in this work indicate that – since the viscoelastic Poisson’s ratio remains relatively constant (slightly increasing) as determined from the confined test at 60 °C—the high strain energy affects the dilatational and deviatoric compliances by roughly the same magnitude, indicating they are similarly impacted by enhanced dissolution rates.