# Laboratory Evaluation of Tensile Creep Behavior of Concrete at Early Ages
## 2 Theoretical Background  
### Constitutive Relations  
The time-dependent behavior of concrete has been effectively explained by the linear viscoelastic theory with the principle of superposition [29]. The general constitutive relations for linear aging viscoelastic materials can be expressed with the superposition integral as follows [30]:  
\[\sigma_{ij}(x_{k},t)=\int\limits_{0}^{t}\,C_{ijkl}(t,tt)\frac{\partial\varepsilon_ {kl}(x_{k},t^{\prime})}{\partial\,tt}dt \tag{1}\]  
where \(\sigma_{ij}\) is the stress tensor, \(\varepsilon_{kl}\) is the strain tensor, \(C_{ijkl}\) is the fourth order tensor of relaxation modulus, \(x_{k}\) is the position vector, \(t\) is the time of interest, and \(t^{\prime}\) is the initial loading time. The relaxation modulus \(C_{ijkl}(t,t^{\prime})\) describes the stress at time \(t\) caused by a unit constant strain applied at time \(t^{\prime}\). Note that the relaxation modulus is a function of the current time, \(t\), and the initial loading time, \(t^{\prime}\). For non-aging materials, it will be a function of the duration of the load (\(t-t^{\prime}\)) only. It is also noted that thermal or shrinkage stresses are not included in Equation (1). Under the linear theory, these environmental stresses may be effectively considered by adding separate terms that consist of relevant mechanical properties relating the environmental effects and corresponding stress developments [5].  
For isotropic materials, the fourth order modulus tensor can be expressed as Equation (2) [30], and, therefore, the constitutive equations reduce to Equation (3) for early-age concrete as an isotropic, linear aging viscoelastic material.  
\[C_{ijkl}(t)=\lambda(t)\delta_{ij}\delta_{kl}+\mu(t)\left(\delta_{ik}\delta_{jl }+\delta_{il}\delta_{jk}\right) \tag{2}\]  
\[\sigma_{ij}(x_{k},t)=\int\limits_{0}^{t}\lambda(t,tt)\,\delta_{ij}\frac{ \partial\varepsilon_{kk}(x_{k},t^{\prime})}{\partial\,tt}dt\,+\,\int\limits_ {0}^{t}\,2\mu(t,tt)\frac{\partial\varepsilon_{ij}(x_{k},t)}{\partial\,tt^{ \prime}}dt \tag{3}\]  
where \(\lambda(t)\) and \(\mu(t)\) are independent relaxation moduli, and \(\delta_{ij}\) is the Kronecker delta. For an effective use of the constitutive equations to early-age concrete, it is crucial to properly define the related material properties, i.e., the time- and age-dependent relaxation modulus functions, \(\lambda(t,t^{\prime})\) and \(\mu(t,t^{\prime})\), for the early-age concrete.  
These general isotropic relaxation moduli can then be expressed in terms of the uniaxial relaxation modulus, \(E(t)\), as shown in Equations (4) and (5). The Poisson's ratio (\(\
u\)) is assumed to be time-independent.  
\[\lambda(t)=\frac{\
u}{(1+\
u)(1-2\
u)}E(t) \tag{4}\]  
\[2\mu(t)=\frac{1}{1+\
u}E(t) \tag{5}\]  
As shown, the relations of the material properties are very similar to those defined in elastic theory. By these relations, the constitutive equation given in Equation (3) can be used for the viscoelastic stress analysis of early-age concrete when the age-dependent uniaxial relaxation modulus, \(E(t,t^{\prime})\), is defined. One way to determine this viscoelastic material property is through a relaxation test [30]. However, due to the difficulties associated with the relaxation test, especially for early-age concrete in tension, a creep test that defines creep compliance is generally preferred. The relaxation modulus can be determined from the compliance by a unique relationship between those two properties.