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**Transformation of curves, surfaces and volume **

Before rigorously deriving the general expressions for transformation of curves, surfaces and volumes, let us obtain these expressions after various simplifying assumptions.

Let us assume the following:

1. The same Cartesian coordinate basis and origin is used to describe the body in both its reference and current configuration.

2. The body is subjected to displacement u and the Lagrangian description of the displacement field is known and is sufficiently smooth for the required derivatives to exist. That is

u = u_{x}(X, Y, Z)e_{x} + u_{y}(X, Y, Z)e_{y} + +u_{z}(X, Y, Z)e_{z}, (3.43)

Figure 3.3: Schematic of the deformation of a line oriented along e_{x} of length âˆ†X.

where u_{x}(X, Y, Z), u_{y}(X, Y, Z) and u_{z}(X, Y, Z) are some known functions of the X, Y and Z the coordinates of the material particles in the reference configuration.

3. It is assumed that the components of the displacement can be approximated as

where the displacement of the material particle occupying the point Xo is u(X_{o}, Y_{o}, Z_{o}) and that displacement of the material particle that is occupying, X1 is u(X_{o} + âˆ†X, Y_{o} + âˆ†Y, Z_{o} + âˆ†Z). Here it is pertinent to point out that the above is a truncated Taylorâ€™s series. Since, we are interested in arbitrarily small values of âˆ†X, âˆ†Y , âˆ†Z, we have truncated the Taylorâ€™s series after the first term. However, this is not a valid approximation for many functions. For example, functions of the form aX^{3/2} , where a is a constant defined over the domain, say 0.01 â‰¤ X â‰¤ 0.1 the value of the derivatives greater then second order would be more than the first order derivative. Hence, the expression derived based on this assumption, though intuitive has its limitations.

First, we are interested in finding how the length of the straight line of length âˆ†X oriented along the e_{x} direction in the reference configuration of the body, as shown in the figure 3.3 has changed. With the above assumptions, the deformed length of the straight line of length âˆ†X oriented along the e_{x} direction in the reference configuration is given by,

where X_{1} = X_{o} + âˆ†Xe_{x}, x_{o} denotes the current position vector of the material particle whose position in the reference configuration is X_{o}, x_{1} denotes the current position vector of the material particle whose position in the reference configuration is X_{1 }and use is made of equation (3.44). Now, if the magnitude of the components of the gradient of the displacement are small (say of the magnitude of 10^{âˆ’3} ), then equation (3.45) could be approximately calculated as,

Hence, the stretch ratio defined as the ratio of the current length to the undeformed length for a line element oriented along the ex direction is,**Figure 3.4: Schematic of the deformation of a face of the cube.**

which can be approximately computed as,

when the components of the displacement gradient are small. Following a similar procedure, one can determine how the length of line elements oriented along any direction changes. We shall derive a general expression for the same later in section 3.6.1. Next we are interested in finding how the area of a face of a small cuboid changes due to deformation. Let us assume that the face of the cuboid whose normal coincides with the e_{z} basis is of interest and the sides of this face of the cuboid are of length âˆ†X along the e_{x} direction and âˆ†Y along the e_{y} direction. Thus, as shown in figure 3.4, X_{o} (= X_{o}e_{x} + Y_{o}e_{y} + Z_{o}e_{z}), X_{1} (= (X_{o}+âˆ†X)e_{x}+Y_{o}e_{y}+Z_{o}e_{z}) and X_{2} (= X_{o}e_{x}+(Y_{o}+âˆ†Y )e_{y}+Z_{o}e_{z}) denote the position vector of the three corners of the face of the cuboid whose normal is ez in the reference configuration and xo, x1 and x2 denote the position of the same three corners of the face of the cuboid in the current configuration. For the same three assumptions listed above, the deformed area of this face of the cuboid is given by

and the orientation of the normal to this deformed face is computed as,

Recollect that in chapter 2 we mentioned that the cross product of two vectors characterizes the area of the parallelogram spanned by them. We have made use of this to obtain the above expressions.

Finally, we find the volume of the deformed cuboid. the box product of three vectors yields the volume of the parallelepiped spanned by them, the deformed volume of the cuboid is given by,

where x_{o}, x_{1}, x_{2} and x_{3} are the position vectors of the material particles in the current configuration corresponding to whose position vector in the reference configuration are

X_{o} (= X_{o}e_{x}+Y_{o}e_{y} +Z_{o}e_{z}), X_{1} (= (X_{o}+âˆ†X)e_{x}+ Y_{o}e_{y} + Z_{o}e_{z}), X_{2 }(= X_{o}e_{x} + (Y_{o} + âˆ†Y )e_{y} + Z_{o}e_{z}) and X_{3 }(= X_{o}e_{x} + Y_{o}e_{y} + (Z_{o} + âˆ†Z)e_{z}) respectively.

When the magnitude of the components of the displacement gradient are small, then the deformed volume of the cuboid could be computed from equation (3.51) as,

by neglecting the quadratic and cubic terms in equation (3.51).

**Transformation of curves **

We know the position vector of any particle belonging to the body at various instances of time, t. But now we are interested in finding how a set of contiguous points forming a curve changes its shape. That is, we are interested in finding how a circle inscribed in the reference configuration changes its shape, say into an ellipse, in the current configuration. As we have required the deformation field to be one to one, closed curves like circle, ellipse, will remain as closed curves and open curves like straight line, parabola remains open. The position vectors of the particles that occupy a curve can be described using a single variable, say Î¾. For example, a circle of radius R in the plane whose normal coincides with e_{z}, as shown in figure 3.5a would be described as (R cos(Î¾), R sin(Î¾), Z_{o}), where R and Z_{o }are constants and 0 â‰¤ Î¾ â‰¤ 2Ï€.

Consider a material curve (or a curve in the reference configuration), **X = Î“(Î¾) âŠ‚ B _{r}**, where Î¾ denotes a parametrization. The material curve by virtue of it being defined in the reference configuration is not a function of time. During a certain motion, the material curve deforms into another curve called the spatial curve, x = Î³(Î¾, t) âŠ‚ B

x = Î³(Î¾, t) = Ï‡(**Î“**(Î¾), t). (3.53)

We denote the tangent vector to the material curve as âˆ†X and the tangent vector to the spatial curve as âˆ†x and are defined by

Figure 3.5: Curves in the reference configuration deforming into another curve in the current configuration

By using (3.53) and the chain rule we find that

Hence, from equation (3.54) and the definition of the deformation gradient, (3.11) we find that

âˆ†**x = Fâˆ†X.** (3.56)

Expression (3.56) clearly defines a linear transformation which generates a vector âˆ†x by the action of the second-order tensor F on the vector âˆ†X. In summary: material tangent vectors map into spatial tangent vectors via the deformation gradient. This is the physical significance of the deformation gradient.

In the literature, the tangent vectors âˆ†x and âˆ†X, in the current and reference configuration are often referred to as the spatial line element and the material line element, respectively. This is correct only when the curve in the reference and current configuration is a straight line.

Next, we introduce the concept of stretch ratio which is defined as the ratio between the current length to its original length. When we say original length, we mean undeformed length, that is the length of the fiber when it is not subjected to any force. It should be pointed out at the outset that many a times the undeformed length would not be available and in these cases it is approximated as the length in the reference configuration.

In general, the stretch ratio depends on the location and orientation of the material fiber for which it is computed. Thus, we can fix the orientation of the material fiber in the reference configuration or in the current configuration. Corresponding to the configuration in which the orientation of the fiber is fixed we have two stretch measures. Recognize that we can fix the orientation of the fibers in only one configuration because its orientation in others is determined by the motion.

First, we consider deformations in which any straight line segments in the reference configuration gets mapped on to a straight line segment in the current configuration. Such a deformation field which maps straight line segments in the reference configuration to straight line segments in the current configuration is called homogeneous deformation. We shall in section 3.10, illustrate that when the deformation is homogeneous, the Cartesian components of the deformation gradient would be a constant. As a consequence of assuming the curve to be a straight line, the tangent and secant to the curve is the same. Hence, saying that we are considering a material fiber^{3} of length âˆ†L initially oriented along **A**, (where **A** is a unit vector) and located at **P**, is same as saying that we are studying the influence of the deformation on the tangent vector, âˆ†L**A** at **P**. Due to some motion of the body, the material particle that occupied the point P will occupy the point p âˆˆ B_{t} and the tangent vector âˆ†LA is going to be mapped on to a tangent vector with length, say âˆ†l and orientation a. From (3.56) we have

âˆ†l**a = âˆ†LFA.** (3.57)

Dotting the above equation with âˆ†la, we get

(âˆ†l)**2** = (âˆ†L)^{2}**FA ï¿½ FA = (âˆ†L) ^{2}F^{t}FA ï¿½ A,** (3.58)

where we have used the fact that a is a unit vector, the relation (3.57) and the definition of transpose. Defining the right Cauchy-Green deformation tensor, C as

and using equation (3.58) we obtain

(3.60)

where Î›

âˆ†L

Dotting the above equation with âˆ†LA, we get

(âˆ†L)

using arguments similar to that used to get (3.58) from (3.57). Defining the left Cauchy-Green deformation tensor or the Finger deformation tensor, B as

B =

^{3}A material fiber is also sometimes called as infinitesimal line elements. |

and using equation (3.62) we obtain

where Î›_{a} represents the stretch ratio of a fiber finally oriented along a.

Next, we would like to compute the final angle between two two straight line segments initially oriented along A_{1} and A_{2}. Recognizing that the angle between two straight line segments, Î± is computed using the expression:

Using (3.65) and similar arguments as that used to obtain the stretch ratio, it can be shown that the final angle, Î±_{f} between the straight line segments initially oriented along A_{1 }and A_{2} is given by

Just, as in the case of stretch ratio, we can also study the initial angle, Î±i between straight line segments that are finally oriented along a_{1} and a_{2}. For this case, following the same steps outlined above we compute

We like to record that as in the case of stretch ratio, Î±_{i} and Î±_{f} depends on the orientation of the line segments. Next, let us see what happens when the deformation is inhomogeneous. For inhomogeneous deformation, the Cartesian components of the deformation gradient would vary spatially and the straight line segment in the reference configuration would have become a curve in the current configuration. Hence, distinction between the tangent and the secant has to be made. Now, the deformed length of the curve, **âˆ†l **corresponding to that of a straight line of length **âˆ†L** oriented along **A** is obtained from equation (3.56) as,

where A is a constant vector, (X(Î¾), Y (Î¾), Z(Î¾)), (X(Î¾), Y (Î¾), Z(Î¾)) denotes the coordinates of the material points that constitutes the straight line and Î¾ varies between 0 and 1 to parameterize the straight line. Thus, for a line oriented along say E_{y} direction of length âˆ†L, starting from a point (X_{o}, Y_{o}, Z_{o}), (see figure 3.6a) X(Î¾) = X_{o}, Y (Î¾) = Y_{o} + Î¾(âˆ†L), Z(Î¾) = Z_{o}. For a line oriented along say (3E_{x} + 4E_{Y} )/5 of length âˆ†L, starting from (X_{o}, Y_{o}, Z_{o}), (see figure 3.6b) X(Î¾) = Xo +3Î¾(âˆ†L)/5, Y (Î¾) = Yo +4Î¾(âˆ†L)/5, Z(Î¾) = Z_{o}. In fact, if one relaxes the assumption that A is a constant in equation (3.68) then the straight line in the reference configuration could be a curve and the deformed length could still be calculated from (3.68).

Hence, the stretch ratio for the case of inhomogeneous deformation is,**Transformation of areas **

Having seen how points, curves, tangent vectors in the reference configuration gets mapped to points, curves and tangent vectors in the current configuration, we are now in a position to look at how surfaces get mapped. Of interest, is how a unit vector, **N** normal to an infinitesimal material surface element âˆ†A map on to a unit vector n normal to the associated infinitesimal spatial surface element âˆ†a.

Let Sr denote the material surface in **B _{r}** that is of interest. Then, an element of area âˆ†A at a point

âˆ†A

Since, âˆ†

âˆ†a

where we have made use of (2.88) and (2.90). Thus, the relation

âˆ†an = det(

called Nansonâ€™s formula is an often used expression in continuum mechanics. Next, we are interested in computing the deformed area, a. For this case, from Nansonâ€™s formula we obtain,

(âˆ†a)

which simplifies to

on using the equation (3.59) and the fact that n is a unit vector. Notice that, once again the deformed area is determined by the right Cauchy-Green deformation tensor.

Next, we seek to obtain the relation between elemental volumes in the reference and current configuration. An element of volume âˆ†V at a interior point

âˆ†v = [âˆ†x, âˆ†y, âˆ†z] = [

where we have used the definition of the determinant (2.77). Since, the coordinate basis vectors for the reference and current configuration have the same handedness, det(

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