## 3 Methods and Results
### D Volume Strain  
Strains deal with the shortening and stretching of nearly linear (straight) objects or lines in a one-dimensional state. One could argue that one-dimensional strain makes no sense because stretching straight lines only changes their length. In other words, a change in form, such as from a circle to an ellipse, can be explained by a change in the length of the lines in different directions. As a result, changing the length of the lines in the sense of strain is appropriate. Figure 3: Types of fissures. (**a**) Styles of fracture deployment: Style I (opening), Style II (sliding), and Style III (tearing); (**b**) fissures expanding in Mode I and effecting from tensile stress; (**c**) fissures expanding in Mode I and effecting from shear stress; (**d**) fissures expanding in Mode I, effecting from compressive stress and lateral expansion. Elongation, stretching, extension, shortening, and contraction are all examples of specific characters that can be used to describe strain values, and they are dimensionless. Line elongation is calculated as e = (L - L0)/L0, where L and L0 are the line lengths after and before deformation, respectively. Elongation in the natural or logarithmic form \(\bar{\text{e}}\) = Ln \(\bar{\text{\epsilon}}\) is also used. In general, positive elongation (e) is used to describe the extensional area, whereas negative extension is referred to as contraction (the concepts of tension and compression are reserved for stress). Line stretching is determined by s = 1 + e, where s is the stretch. As a result, s = L/L0 stretching factors are commonly associated with the structural analysis of extensional basins and rifts. These are sometimes referred to as b-factors, but they are identical to s.  
Quadratic elongation, \(\lambda\) = S2, is equal to the Eigenvalues of the matrix (D). Quadratic stretch is a more accurate concept, as it considers the square of the stretch, rather than its length. Strain is affected by changes in an element's shape and dimensions. When the deformation is along a single axis, strain concepts can be defined as follows:  
\[\lambda^{\prime} =1/\lambda \tag{1}\] \[\text{R}_{\text{XY}} =\text{R}.\text{cosy}\] (2) \[\text{R}_{\text{YZ}} =\text{R}.\text{cos}\alpha\] (3) \[\text{R}_{\text{XZ}} =\text{R}.\text{cos}\beta\] (4) \[\text{D} =((\text{R}_{\text{XY}}-1)^{2}+(\text{R}_{\text{YZ}}-1)^{2})^{1/2} \tag{5}\]  
where \(\alpha\), B, and \(\gamma\) are the angles formed by the normal of the plane and each of the principal stress axes (x, y, z). The normal strains are generally parallel to three perpendicular axes. Typically, the stress function modifies the element's volume. For this reason, volumetric strain (e) is calculated as a function of normal stresses:  
\[\text{e}=\frac{\text{change volume}}{\text{original volume}} \tag{6}\]  
The normal strains similarly affect the volume strains. The original volume of the rectangular solid was calculated using the formula V0 = Ax. \(\Delta\)y.\(\Delta\)z. (Figure 4), whereas for a deformed body, it would be calculated as:  
\[\text{V}=\Delta\text{x}(1+\varepsilon_{\text{x}})\times\Delta\text{y}(1+ \varepsilon_{\text{y}})\times\Delta\text{z}(1+\varepsilon_{\text{z}})= \Delta\text{x}\Delta\text{y}\Delta\text{z}(1+\varepsilon_{\text{x}}+ \varepsilon_{\text{y}}+\varepsilon_{\text{z}}+\varepsilon_{\text{x}} \varepsilon_{\text{y}}+\varepsilon_{\text{y}}\varepsilon_{\text{z}}+\] (7) \[\varepsilon_{\text{z}}\varepsilon_{\text{x}}+\varepsilon_{\text{x}} \varepsilon_{\text{y}}\varepsilon_{\text{z}})\] Since the strains are small, the high order terms are ignored, where \[\text{V}\approx\Delta\text{x}\Delta\text{y}\Delta\text{z}(1+\varepsilon_{ \text{x}}+\varepsilon_{\text{y}}+\varepsilon_{\text{z}})\] (8) Finally, volume strain can be defined as:  
\[\lambda^{\prime} =1/\lambda \tag{1}\]0 (a) Coordinate axes in a block diagram; (b) coordinate strain axes of the block for the obtained volume strain. \[\lambda^{\prime} =1/\lambda \tag{1}\]1