*2.2.1 Relaxation tests*

Relaxation is defined as stress (or load) decrease over time when the deformation (or strain) is kept constant. Commonly, the axis on which the stress is applied (i.e. axial stress used) determines the deformation's axes that are maintained constant (i.e. axial strain – constant). It has been observed that relaxation behavior is related not only to time-dependent phenomena like creep but also to timedependent damage evolution of new or pre-existing cracks growth and evolution in the specimen that initiates during loading [9, 32, 33].

**Figure 5** shows the stages during a stress relaxation test from A to C. The rock is initially loaded in the axial direction up to point A, which is considered the strain threshold at which the applied strain is held constant (points A to C). In this regard, these tests are often referred to as strain-controlled. Overtime, existing cracks and/ or new cracks are formed and propagated at this strain threshold, contributing to the observed stress decrease (relaxation). When this stress relaxation reaches an asymptote (no further decrease is observed), the test is terminated, which implies that crack growth stabilization is achieved [19].

It should be stated that suggested standard test guidelines on relaxation tests on rock samples are not provided by ISRM. However, there are guidelines provided by ASTM [34] for relaxation testing performed on man-made materials and structures. In section 3.2 this standard has been adopted and adjusted for rock relaxation testing.

**Figure 5.** *Relaxation test: (a) stress–strain response, (b) strain- time response, (c) stress-time response.* *Time-Dependent Behavior of Rock Materials DOI: http://dx.doi.org/10.5772/intechopen.96997*

**Figure 6.**

*Static load test: (a) stress–strain response, (b) stress-time response, (c) strain-time response.*

## *2.2.2 Static load tests*

To investigate creep, this time-dependent deformation of materials subjected to constant load or stress less than its short-term strength, static load tests are performed. In materials, here is a minimum load or stress, which enables them to undergo creep behavior, below which no creep is observed [9, 35, 36]. Elevated differential stress triggers the deformation of crystal lattices, leading to straining of the minerals, potentially microcracking, and eventually measurable strain of the rock element. **Figure 6** presents a typical stress–strain-time response of a (uniaxial) creep test. The rock sample is loaded until point A, the stress threshold where it is held constant. Over time the strain increases at different rates up to point B, where failure occurs. This test is usually referred to as load-controlled or stresscontrolled tests.

Failure of the specimen usually denoted the completion of the test. However, many static load tests are terminated when a constant strain-rate is achieved, inferring the transition to the secondary stage of creep. For static load tests, ISRM [37] has suggested standard guidelines.

### **2.3 Time-dependent models**

The time-dependent mechanisms are usually investigated by developing analytical methods adopting rheological models (comprising mechanical analogues) and empirical models based on laboratory testing data. Specifically, creep behavior is mathematically represented by the Burgers model. This model combines two simplified linear visco-elastic mechanical analogues in series: the Kelvin and the Maxwell that simulate a delayed manifestation of a static response due to boundary conditions alteration and a continued strain rate relaxation overtime under static boundary conditions, respectively shown in **Figure 7**.

Deformation that occurs at constant loading condition through time can be expressed using Eq. (1) [38], where: ε1 is the axial strain, σ1 is the constant axial stress, K is the bulk modulus, ηK is Kelvin's model viscosity, ηM is Maxwell's model viscosity, GK is Kelvin's shear modulus, GM is Maxwell's shear modulus. ηK, ηM, GK, GM are the visco-elastic parameters and are considered properties of the rock.

$$\mathcal{L}\_1(t) = \frac{2\sigma\_1}{\Re K} + \frac{\sigma\_1}{\Re G\_M} + \frac{\sigma\_1}{\Re G\_K} - \frac{\sigma\_1}{\Re G\_K} e^{-\left(\frac{\overline{G\_K}}{\eta\_K}t\right)}\tag{1}$$

During stress relaxation, the strain-state is controlled and remains constant, thus rearranging Eq. (1) for a constant strain component, the material's stress state is changing according to Eq. (2).

**Figure 7.**

*Idealized creep and relaxation behavioral curves and the equivalent visco-elastic components in the Burgers model.*

$$\sigma\_1 \left( t \right) = \varepsilon\_1 \left[ G\_M e^{-\left(\frac{G\_M}{\eta\_M}t\right)} + G\_K e^{-\left(\frac{G\_K}{\eta\_K}t\right)} \right] \tag{2}$$

Goodman's [38] approach is usually adopted to derive the Bugers model parameters by curve fitting laboratory creep testing results. Using a similar approach for determining parameters and assuming that the material's behavior can be represented by the linear visco-elastic Burgers body in unconfined compression [33] found that the same parameters (i.e. viscosities and shear moduli) can be also derived from stress relaxation tests, (**Figure 7**).

In reality and embedded in this mathematical concept are the three stages of creep that follow the instantaneous response (0th stage) to changed boundary conditions resulting to a constant stress-state as follows:

• 1st stage or primary or transient creep where the delayed adjustment to a new equilibrium state takes place through visco-elastic (reversible) deformation, and may be accompanied by some irreversible behavior, resulting in strain accumulation with decreasing rate over time. This stage is commonly simulated with the Kelvin model analogue.

## *Time-Dependent Behavior of Rock Materials DOI: http://dx.doi.org/10.5772/intechopen.96997*


A combination of Kelvin and Maxwell model components is referred to as the Burgers model which can be used to simulate stages 1 and 2 in combination.
