We now proceed to state some notions that are commonly used in most approaches of Elasticity theory (Chandrasekharaiah and Debnath, 1994). These will later be used in the analysis on which the unwrapping of the parasite is based.
Definition of the Stress and the Strain Tensor
Consider a one-to-one correspondence between all points of the deformed and the undeformed parasite states. Thus, let us consider two arbitrary points, say and of an undeformed parasite element and let and be their unique images in the deformed parasite body. In other words, due to the deformation, point moves to and point to . We consider the lengths of and : and
Now one defines the relative elongations along the x- and y-axes:
Along the x-axis, we set , in which case, and above become and , respectively. Thus, the relative elongation is defined to be (1)
Similarly, along the y-axis we set and thus the relative elongation εyy is defined to be: (2)
If the relative elongations are small, after expanding the square roots of (1) and (2) in Taylor series, we obtain
Moreover, using the same assumption, the tangent of the angle of deformation of the x-axis, , and the y-axis,, become and respectively (Figure1). Adopting the hypothesis of small relative elongations, it results that , implying that the initially right angle is deformed by (4)
Element differential deformation
Therefore, after using the shear stress definition , we adopt the standard strain tensor definition: .
Next, in order to study the elastic forces’ distribution throughout the parasite’s body, we proceed by considering an arbitrary differential element in the parasite body, starting at point with vertices, , , (Figure2). Let the force per unit area/length acting on the side be , where the first subscript denotes the axis to which the side is vertical, while the second subscript denotes the vector component axis. Similarly, . It is evident that the four functions , , , , suffice to determine the stress condition of the differential element under consideration. Hence, we define the standard stress tensor .
Differential element strain forces