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\mathbf {a} = \mathbf {C} \cdot \mathbf {A}.
Variations ±a in this field are obtained as
\delta \mathbf {a} = \delta \mathbf {C} \cdot \mathbf {A},
where ±C is the linearized rotation matrix; that is, the variation of the orthogonal tensor C. On the other hand, the variation can be defined in terms of the linearized rotation field ±µ:
\delta \mathbf {a} = \delta \boldsymbol {\theta} \times \mathbf {a} = \widehat {\delta \boldsymbol {\theta}} \cdot \mathbf {a} = \widehat {\delta \boldsymbol {\theta}} \cdot \mathbf {C} \cdot \mathbf {A}.
Consequently, it follows that
\delta \mathbf {C} = \widehat {\delta \pmb {\theta}} \cdot \mathbf {C}.
It is important to note that the linearized rotation ±µ, which is analogous to the angular velocity in dynamics, is not the variation of the rotation vector Á. By a straightforward (but involved) calculation, it can be shown that the variation of the rotation vector ( ±Á) is related to the linearized rotation ±µ by
Equation 1.3.1-6
\delta \boldsymbol {\theta} = \mathbf {H} (\boldsymbol {\phi}) \cdot \delta \boldsymbol {\phi},
where
\mathbf {H} (\phi) = \frac {1}{\| \phi \| ^ {2}} \phi \phi + \frac {\sin \| \phi \|}{\| \phi \|} \left[ \mathbf {I} - \frac {1}{\| \phi \| ^ {2}} \phi \phi \right] + \frac {1 - \cos \| \phi \|}{\| \phi \| ^ {2}} \hat {\phi}.
The inverse of H(Á) is
\mathbf {H} (\phi) ^ {- 1} = \frac {1}{\| \phi \| ^ {2}} \phi \phi + \frac {\| \phi \| \sin \| \phi \|}{2 (1 - \cos \| \phi \|)} \left[ \mathbf {I} - \frac {1}{\| \phi \| ^ {2}} \phi \phi \right] - \frac {1}{2} \hat {\phi}.
Let dµ represent an infinitesimal change in the rotation field. A direct calculation of the variation of dµ, which is equivalent to calculation of the second variation of either C or a, leads to an expression that is not symmetric in the variations ±µ and the changes dµ. However, it is shown in Simo (1992) that the correct definition of the Hessian operator--that is, the "covariant" derivative of the weak form of the balance equations--requires only the symmetric part (with respect to the variations) of the second variation. Thus, without loss of generality, we can write
\mathrm{d} (\delta \mathbf {C}) = \frac {1}{2} (\widehat {\delta \pmb {\theta}} \cdot \widehat {\mathrm{d} \pmb {\theta}} \cdot \mathbf {C} + \widehat {\mathrm{d} \pmb {\theta}} \cdot \widehat {\delta \pmb {\theta}} \cdot \mathbf {C}).
Similarly, the second variation of the vector field rotated by C can be written as
\mathrm{d} (\delta \mathbf {a}) = \frac {1}{2} \left[ \delta \pmb {\theta} \times (\mathrm{d} \pmb {\theta} \times \mathbf {a}) + \mathrm{d} \pmb {\theta} \times (\delta \pmb {\theta} \times \mathbf {a}) \right]
= - (\delta \pmb {\theta} \cdot \mathrm{d} \pmb {\theta}) \mathbf {a} + \frac {1}{2} \left[ (\delta \pmb {\theta} \cdot \mathbf {a}) \mathrm{d} \pmb {\theta} + (\mathrm{d} \pmb {\theta} \cdot \mathbf {a}) \delta \pmb {\theta} \right].
Velocity and acceleration
Taking the time derivative of the rotation matrix, we find with the same arguments as used in the calculation of the variations that
\dot {\mathbf {C}} = \widehat {\boldsymbol {\omega}} \cdot \mathbf {C},
\ddot {\mathbf {C}} = \widehat {\dot {\boldsymbol {\omega}}} \cdot \mathbf {C} + \widehat {\boldsymbol {\omega}} \cdot \widehat {\boldsymbol {\omega}} \cdot \mathbf {C},
where \widehat { \omega } is the angular velocity matrix. Equivalently, the first and second time derivative of a are written as
\dot {\mathbf {a}} = \boldsymbol {\omega} \times \mathbf {a},
\ddot {\mathbf {a}} = \dot {\boldsymbol {\omega}} \times \mathbf {a} + \boldsymbol {\omega} \times (\boldsymbol {\omega} \times \mathbf {a}).
The instantaneous angular velocity vector ! is related to the time rate of change of the rotation vector by the relation
\pmb {\omega} = \mathbf {H} (\pmb {\phi}) \cdot \dot {\pmb {\phi}},
where H(Á) is given by Equation 1.3.1-6.
In the linearization of the dynamic balance equations, it is necessary to calculate the variation of the angular velocity, d!. This quantity, however, can be calculated only by linearizing the specific algorithm used for the time integration of the dynamic equations.
Coupling of rotations: constant velocity joint
Next, a more rigorous treatment of the two-dimensional constant velocity joint described in ``MPC,'' Section 23.2.13 of the ABAQUS/Standard User's Manual, is presented. This derivation exemplifies some of the issues associated with the treatment of finite rotations. ``Uniform collapse of straight and curved pipe segments,'' Section 1.1.5 of the ABAQUS Benchmarks Manual, deals with a different finite rotation constraint and tackles additional complications.
Let a, b, c (see Figure 1.3.1-1) be the nodes making up the joint, with a the dependent node.
Figure 1.3.1-1 Nonlinear MPC example--constant velocity joint.
text_image
y x φ^c c b a φ^b
The joint is operated by prescribing an axial rotation { \boldsymbol { \phi } } ^ { c } = \phi ^ { c } \mathbf { e } _ { x } at c and an out-of-plane rotation \boldsymbol { \phi } ^ { b } = \phi ^ { b } \mathbf { e } _ { z } at b. The compounding of these two prescribed rotation fields will determine the total rotation at a. We can formally write this constraint as follows:
\mathbf {f} (\pmb {\phi} ^ {a}, \pmb {\phi} ^ {b}, \pmb {\phi} ^ {c}) = \pmb {\phi} ^ {a} - \pmb {\phi} ^ {b} \circ \pmb {\phi} ^ {c} = \mathbf {0}.
The constraint can be written in terms of the rotation matrices as
Equation 1.3.1-7
\mathbf {C} (\pmb {\phi} ^ {a}) - \mathbf {C} (\pmb {\phi} ^ {b}) \cdot \mathbf {C} (\pmb {\phi} ^ {c}) = \mathbf {0}.
With the previously defined variational expressions, the constraint can be linearized as
\widehat {\delta \pmb {\theta} ^ {a}} \cdot \mathbf {C} (\pmb {\phi} ^ {a}) - \widehat {\delta \pmb {\theta} ^ {b}} \cdot \mathbf {C} (\pmb {\phi} ^ {b}) \cdot \mathbf {C} (\pmb {\phi} ^ {c}) - \mathbf {C} (\pmb {\phi} ^ {b}) \cdot \widehat {\delta \pmb {\theta} ^ {c}} \cdot \mathbf {C} (\pmb {\phi} ^ {c}) = \mathbf {0}.
This expression can be simplified by right-multiplying the expression by { \bf C } ^ { T } ( \phi ^ { a } ) and by making use of the constraint Equation 1.3.1-7, which yields
\widehat {\delta \pmb {\theta} ^ {a}} - \widehat {\delta \pmb {\theta} ^ {b}} - \mathbf {C} (\pmb {\phi} ^ {b}) \cdot \widehat {\delta \pmb {\theta} ^ {c}} \cdot \mathbf {C} ^ {T} (\pmb {\phi} ^ {b}) = \mathbf {0},
which can be written in vector form as
\delta \pmb {\theta} ^ {a} - \delta \pmb {\theta} ^ {b} - \mathbf {C} (\pmb {\phi} ^ {b}) \cdot \delta \pmb {\theta} ^ {c} = \mathbf {0}.
Since
\mathbf {C} (\pmb {\phi} ^ {b}) = \left( \begin{array}{c c c} \cos \phi^ {b} & - \sin \phi^ {b} & 0 \\ \sin \phi^ {b} & \cos \phi^ {b} & 0 \\ 0 & 0 & 1 \end{array} \right),
the linearized constraint is indeed identical to the one derived based on simple linear considerations in the ABAQUS/Standard User's Manual.
The linearized constraint is used for the calculation of equilibrium. It can also be used for the recovery of the dependent rotation, \phi ^ { a } , as was done in the ABAQUS/Standard User's Manual. The resulting rotation will satisfy the constraint approximately (unless one of the angles \phi ^ { b } or \phi ^ { c } is constant, in which case the constraint is linear and the recovery is exact).
For an exact enforcement of the constraint, user subroutine MPC must define the components of the total rotation vector \phi ^ { a } exactly. To do so, \phi ^ { a } must be updated based on the current values of \phi ^ { b } and \phi ^ { c } . This is most easily accomplished with the aid of the quaternion parameters. Let
\pmb { q } ^ { b } = ( \cos ( \phi ^ { b } / 2 ) , \sin ( \phi ^ { b } / 2 ) \mathbf { e } _ { z } ) and { \pmb q } ^ { c } = ( \cos ( \phi ^ { c } / 2 ) ; sin ( \phi ^ { c } / 2 ) \mathbf { e } _ { x } ) be the quaternion
parameterizations associated with the finite rotation vectors \phi ^ { b } and \phi ^ { c } , respectively. The total compound rotation \phi ^ { a } is given by the quaternion { \pmb q } ^ { a } = ( q _ { 0 } ^ { a } , { \bf q } ^ { a } ) , where
q _ {0} ^ {a} = \cos (\phi^ {b} / 2) \cos (\phi^ {c} / 2),
\mathbf {q} ^ {a} = \cos (\phi^ {b} / 2) \sin (\phi^ {c} / 2) \mathbf {e} _ {x} + \sin (\phi^ {b} / 2) \sin (\phi^ {c} / 2) \mathbf {e} _ {y} + \cos (\phi^ {b} / 2) \sin (\phi^ {c} / 2) \mathbf {e} _ {z},
according to the quaternion compound formula Equation 1.3.1-4. The rotation vector \phi ^ { a } is extracted from the quaternion { \pmb q } ^ { a } as follows:
\pmb {\phi} ^ {a} = \phi^ {a} \frac {\mathbf {q} ^ {a}}{\| \mathbf {q} ^ {a} \|} \quad \mathrm{with} \quad \phi^ {a} = 2 \tan^ {- 1} \left[ \frac {\| \mathbf {q} ^ {a} \|}{q _ {0} ^ {a}} \right],
where \left\| \mathbf { q } ^ { a } \right\| is the norm of the vector { \bf q } ^ { a } .
``MPC,'' Section 23.2.13 of the ABAQUS/Standard User's Manual, shows the implementation of the linearized form of the constraint in user subroutine MPC. The implementation of the exact nonlinear constraint is shown below:
SUBROUTINE MPC (UE, A, JDOF, MDOF, N, JTYPE, X, U, UINIT, MAXDOF, LMPC, * KSTEP, KINC, TIME, NT, NF, TEMP, FIELD)
C
INCLUDE 'ABA_PARAM.INC'
C
DIMENSION UE (MDOF), A (MDOF, MDOF, N), JDOF (MDOF, N), X (6, N), * U (MAXDOF, N), UINIT (MAXDOF, N), TIME (2), TEMP (NT, N), * FIELD (NF, NT, N)
PARAMETER (SMALL = 1.E-14)
C
IF (JTYPE.EQ.1) THEN
A(1,1,1) = 1.
A(2,2,1) = 1.
A(3,3,1) = 1.
A(3,1,2) = -1.
A(1,1,3) = -COS(U(6,2))
A(2,1,3) = -SIN(U(6,2))
C
JDOF(1,1) = 4
JDOF(2,1) = 5
JDOF(3,1) = 6
JDOF(1,2) = 6
JDOF(1,3) = 4
C
CPHIB = COS(0.5*U(6,2))
SPHIB = SIN(0.5*U(6,2))
CPHIC = COS(0.5*U(4,3))
SPHIC = SIN(0.5*U(4,3))
C
QA0 = CPHIB*CPHIC
QAX = CPHIB*SPHIC
QAY = SPHIB*SPHIC
QAZ = CPHIB*SPHIC
C
QAMAG = SQRT(QAX*QAX + QAY*QAY + QAZ*QAZ)
IF (QAMAG .GT. SMALL) THEN
PHIA = 2.*ATAN2(QAMAG, QA0)
UE(1) = PHIA*QAX/QAMAG
UE(2) = PHIA*QAY/QAMAG
UE(3) = PHIA*QAZ/QAMAG
ELSE
UE(1) = 0.
UE(2) = 0.
UE(3) = 0.
END IF
END IF
C
RETURN
END
1.4 Deformation, strain, and strain rates
1.4.1 Deformation
In any structural problem the analyst describes the initial configuration of the structure and is interested in its deformation throughout the history of loading. The material particle initially located at some position X in space will move to a new position x: since we assume material cannot appear or disappear, there will be a one-to-one correspondence between x and X, so we can always write the history of the location of a particle as
Equation 1.4.1-1
\mathbf {x} = \mathbf {x} (\mathbf {X}, t)
and this relationship can be inverted--we know X when we know x and t. Now consider two neighboring particles, located at X and at X + dX in the initial configuration. In the current configuration we must have
Equation 1.4.1-2
d \mathbf {x} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}} \cdot d \mathbf {X}
using the "mapping" Equation 1.4.1-1.
The matrix
Equation 1.4.1-3
\mathbf {F} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}}
is called the deformation gradient matrix, and Equation 1.4.1-2 is written
Equation 1.4.1-4
d \mathbf {x} = \mathbf {F} \cdot d \mathbf {X}.
As the material behavior depends on the straining of the material and not on its rigid body motion, those parts of the motion in the vicinity of a material point must be distinguished. Looking at an infinitesimal gauge length dX emanating from the particle initially at X, we can measure its initial and current lengths as
d L ^ {2} = d \mathbf {X} ^ {T} \cdot d \mathbf {X} \quad \mathrm{and} \quad d l ^ {2} = d \mathbf {x} ^ {T} \cdot d \mathbf {x},
so the "stretch ratio" of this gauge length is
Equation 1.4.1-5
\lambda = \frac {d l}{d L} = \sqrt {\frac {d \mathbf {x} ^ {T} \cdot d \mathbf {x}}{d \mathbf {X} ^ {T} \cdot d \mathbf {X}}}.
If ¸ = 1, there is no strain of this infinitesimal gauge length--it has undergone rigid body motion only.
Now using Equation 1.4.1-4,
d \mathbf {x} ^ {T} \cdot d \mathbf {x} = d \mathbf {X} ^ {T} \cdot \mathbf {F} ^ {T} \cdot \mathbf {F} \cdot d \mathbf {X},
so that, from Equation 1.4.1-5,
Equation 1.4.1-6
\begin{array}{l} \lambda^ {2} = \frac {d \mathbf {X} ^ {T}}{\sqrt {d \mathbf {X} ^ {T} \cdot d \mathbf {X}}} \cdot \mathbf {F} ^ {T} \cdot \mathbf {F} \cdot \frac {d \mathbf {X}}{\sqrt {d \mathbf {X} ^ {T} \cdot d \mathbf {X}}} \\ = \mathbf {N} ^ {T} \cdot \mathbf {F} ^ {T} \cdot \mathbf {F} \cdot \mathbf {N}, \\ \end{array}
where N is a unit vector in the direction of the gauge length d \mathbf { X }
Equation 1.4.1-6 shows how to measure the stretch ratio associated with any direction, N, at any material point defined by X (or by x). Useful results are obtained when we vary the direction defined by N at a particular material point and look for stationary values of the stretch ratio, ¸. Since N must always be a unit vector, stationary values of \lambda ^ { 2 } are obtained by solving the constrained variational equation
\delta \left\{\mathbf {N} ^ {T} \cdot \mathbf {F} ^ {T} \cdot \mathbf {F} \cdot \mathbf {N} - p \left(\mathbf {N} ^ {T} \cdot \mathbf {N} - 1\right) \right\} = 0,
where p is a Lagrange multiplier, introduced to retain the constraint
\mathbf {N} ^ {T} \cdot \mathbf {N} = 1.
Taking the variation gives back the constraint (conjugate to \delta p ) and, conjugate to ±N, gives
Equation 1.4.1-7
(\mathbf {F} ^ {T} \cdot \mathbf {F} - p \mathbf {I}) \cdot \mathbf {N} = 0.
Taking the dot product of the left-hand side of this equation with N and comparing with Equation 1.4.1-6 identifies p = \lambda ^ { 2 } , so Equation 1.4.1-7 is
Equation 1.4.1-8
\left(\mathbf {F} ^ {T} \cdot \mathbf {F} - \lambda^ {2} \mathbf {I}\right) \cdot \mathbf {N} = 0.
This problem is an eigenvalue one that can be solved for the three extreme values of \lambda ^ { 2 } . Since ¸ is always real and positive (and nonzero), \lambda ^ { 2 } > 0 , and hence \mathbf { F } ^ { T } \cdot \mathbf { F } must be positive definite. Equation 1.4.1-8 thus gives three real, positive eigenvalues, \lambda _ { I } , \lambda _ { I I } , \lambda _ { I I I } , the "principal stretches," with three corresponding eigenvectors, \mathbf { N } _ { I } , \mathbf { N } _ { I I } , \mathbf { N } _ { I I I } , which will be orthogonal, by Equation 1.4.1-8, if the corresponding eigenvalues are different, and can be orthogonalized otherwise. The { \bf N } _ { I } are the principal directions of strain.
Now let \mathbf { n } _ { I } , \mathbf { n } _ { I I } , \mathbf { n } _ { I I I } be unit vectors corresponding to \mathbf { N } _ { I } , \mathbf { N } _ { I I } , \mathbf { N } _ { I I I } , but in the current configuration, so that, using Equation 1.4.1-4,
\mathbf {n} _ {I} = \frac {1}{\lambda_ {I}} \mathbf {F} \cdot \mathbf {N} _ {I}, \mathrm{etc.}
Then
\begin{array}{l} \mathbf {n} _ {I} ^ {T} \cdot \mathbf {n} _ {I I} = \frac {\hat {\mathbf {\lambda}}}{\lambda_ {I} \lambda_ {I I}} \mathbf {N} _ {I} ^ {T} \cdot \mathbf {F} ^ {T} \cdot \mathbf {F} \cdot \mathbf {N} _ {I I} \\ = \frac {1}{\lambda_ {I} \lambda_ {I I}} \lambda_ {I I} ^ {2} \mathbf {N} _ {I} ^ {T} \cdot \mathbf {N} _ {I I} \\ = 0 \\ \end{array}
by the orthogonality results just mentioned. Thus, \mathbf { n } _ { I } , \mathbf { n } _ { I I } , and { \bf n } _ { I I I } will also be an orthogonal set. Since each is a unit vector,
\mathbf {n} _ {I} = \mathbf {R} \cdot \mathbf {N} _ {I},
\mathbf {n} _ {I I} = \mathbf {R} \cdot \mathbf {N} _ {I I},
\mathbf {n} _ {I I I} = \mathbf {R} \cdot \mathbf {N} _ {I I I},
where R is the same pure rigid body rotation matrix in each of these equations. A pure rigid body motion matrix has the property that its inverse is its transpose: { \bf R } ^ { T } = { \bf R } ^ { - 1 } . Comparing the principal stretch directions in the current and original configurations, therefore, isolates the rigid body rotation and the stretch. Finding the principal stretch ratios and their directions thus provides one solution to the problem of isolating straining motion and rigid body motion in the vicinity of a material point.
Now consider a gauge length in the reference configuration, d \mathbf { X } _ { I } , directed along { \bf N } _ { I } . The same infinitesimal material line in the current configuration will be along { \bf n } _ { I } and will be stretched by \lambda _ { I } , so that
d \mathbf {x} _ {I} = \lambda_ {I} \mathbf {R} \cdot d \mathbf {X} _ {I}.
Similarly, along the other principal directions,
d \mathbf {x} _ {I I} = \lambda_ {I I} \mathbf {R} \cdot d \mathbf {X} _ {I I}
and
d \mathbf {x} _ {I I I} = \lambda_ {I I I} \mathbf {R} \cdot d \mathbf {X} _ {I I I}.
Since ( \mathbf { N } _ { I } , \mathbf { N } _ { I I } , \mathbf { N } _ { I I I } ) is an orthonormal set of base vectors in the reference configuration, any infinitesimal material line (gauge length) dX at X can be written in terms of its components in this basis:
d \mathbf {X} = d \mathbf {X} _ {I} + d \mathbf {X} _ {I I} + d \mathbf {X} _ {I I I},
where
d \mathbf {X} _ {I} = \mathbf {N} _ {I} \mathbf {N} _ {I} ^ {T} \cdot d \mathbf {X}, \quad \mathrm{etc.}
Each of the vectors d { \mathbf { X } } _ { I } moves and stretches to the corresponding d { \bf x } _ { I } , , as defined above. Thus, the current gauge length, dx, is
\begin{array}{l} d \mathbf {x} = d \mathbf {x} _ {I} + d \mathbf {x} _ {I I} + d \mathbf {x} _ {I I I} \\ = \lambda_ {I} \mathbf {R} \cdot d \mathbf {X} _ {I} + \lambda_ {I I} \mathbf {R} \cdot d \mathbf {X} _ {I I} + \lambda_ {I I I} \mathbf {R} \cdot d \mathbf {X} _ {I I I} \\ = \left(\lambda_ {I} \mathbf {R} \cdot \mathbf {N} _ {I} \mathbf {N} _ {I} ^ {T} + \lambda_ {I I} \mathbf {R} \cdot \mathbf {N} _ {I I} \mathbf {N} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {R} \cdot \mathbf {N} _ {I I I} \mathbf {N} _ {I I I} ^ {T}\right) \cdot d \mathbf {X} \\ = \left(\lambda_ {I} \mathbf {n} _ {I} \mathbf {N} _ {I} ^ {T} + \lambda_ {I I} \mathbf {n} _ {I I} \mathbf {N} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {n} _ {I I I} \mathbf {N} _ {I I I} ^ {T}\right) \cdot d \mathbf {X} \\ = \left(\lambda_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \lambda_ {I I} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}\right) \cdot \mathbf {R} \cdot d \mathbf {X} \\ \end{array}
which we write as
Equation 1.4.1-9
d \mathbf {x} = \mathbf {V} \cdot \mathbf {R} \cdot d \mathbf {X},
where
Equation 1.4.1-10
\mathbf {V} = \left(\lambda_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \lambda_ {I I} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}\right)
is the "left stretch" matrix, which is the sum of three dyadic products.
Comparison with the definition of the deformation gradient, Equation 1.4.1-4, shows that
Equation 1.4.1-11
\mathbf {F} = \mathbf {V} \cdot \mathbf {R},
which is the polar decomposition theorem--that any motion can be represented as a pure rigid body rotation, followed by a pure stretch of three orthogonal directions. The polar decomposition theorem is important because it allows us to distinguish the straining part of the motion from the rigid body rotation. Specifically, F completely defines the relative motions of material particles in the infinitesimal neighborhood of the material particle that was at X in the reference configuration; and the left stretch matrix, V, completely defines the deformation of the material particles at X. The rotation matrix R defines the rigid body rotation of the principal directions of strain ( \mathbf { N } _ { I } in the reference configuration; \mathbf { n } _ { I } in the current configuration). R represents only the rigid body rotation of the material at the point under consideration in some average sense: in a general motion, each infinitesimal gauge length emanating from a material particle has a different amount of rotation. This distinction between the rotation of the principal directions of strain, R, and the rotations of individual directions in the material becomes significant when we must discuss large deformations of nonisotropic materials. Nevertheless, we have established an important result: if F = R only, we know there is no deformation of the material in the immediate neighborhood of the point originally at X and currently at x , since in this case V = I and so \lambda _ { I } = \lambda _ { I I } = \lambda _ { I I I } = 1 . \mathbf { V } - \mathbf { I } must be nonzero for there to be any deformation of the material at the point in question: in this sense V ¡ I (and, hence, V itself) is sufficient to define the deforming part of the motion (it contains complete information about all except pure rigid body rotation of the point). For this reason--so that, later in the development, we will be able to link the kinematics to the stressing of the material--we will need to be
able to isolate V from F. It is easy to obtain V ¢ V, for
\begin{array}{l} \mathbf {F} \cdot \mathbf {F} ^ {T} = \mathbf {V} \cdot \mathbf {R} \cdot \mathbf {R} ^ {T} \cdot \mathbf {V} ^ {T} \\ = \mathbf {V} \cdot \mathbf {V}, \\ \end{array}
since { \bf R } ^ { T } = { \bf R } ^ { - 1 } and V is symmetric.
Since we originally defined V from the principal stretches and their principal directions in the current configuration as
\mathbf {V} = \lambda_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \lambda_ {I I} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \lambda_ {I I I} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T},
then
Equation 1.4.1-12
\mathbf {F} \cdot \mathbf {F} ^ {T} = \mathbf {V} \cdot \mathbf {V} = \lambda_ {I} ^ {2} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \lambda_ {I I} ^ {2} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \lambda_ {I I I} ^ {2} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}.
We see that the eigenvalues of \mathbf { F } \cdot \mathbf { F } ^ { T } , are { \lambda _ { I } } ^ { 2 } , { \lambda _ { I I } } ^ { 2 } , and { \lambda _ { I I I } } ^ { 2 } , and the corresponding eigenvectors are \mathbf { n } _ { I } , \mathbf { n } _ { I I } , and { \bf n } _ { I I I } . We can then construct V. The deformation at the point is, thus, readily obtained by multiplying a 3 \times 3 matrix with its transpose ( \mathbf { F } \cdot \mathbf { F } ^ { T } ) and solving the real eigenproblem for the resulting (symmetric) matrix. We can then obtain the rotation R as
\mathbf {R} = \mathbf {V} ^ {- 1} \cdot \mathbf {F}.
Since V has been constructed from its eigenvalues and eigenvectors, its inverse is immediately available:
\mathbf {V} ^ {- 1} = \frac {1}{\lambda_ {I}} \mathbf {n} _ {I} \mathbf {n} _ {I} ^ {T} + \frac {1}{\lambda_ {I I}} \mathbf {n} _ {I I} \mathbf {n} _ {I I} ^ {T} + \frac {1}{\lambda_ {I I I}} \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} ^ {T}.
So far we have written the results quite generally, without reference to any particular coordinate system. To perform computations we must choose a basis system to express these results as arrays of individual numbers. We now do so with some generality with respect to the choice of basis system. The justification for retaining generality at this stage is twofold: as an exercise, to provide a little more familiarity in the notation system we have chosen to use in this manual, and because we do need some--but, as it turns out, not all--of the generality when we have to deal with shell elements, where it is undesirable to use the rectangular Cartesian base vectors of the global, spatial system because the natural orientation of the shell reference surface causes us to prefer to choose two of the base vectors to be tangent to the shell's reference surface and the other to be normal to this surface. This preference causes us to need two basis systems: one associated with the body in its current configuration, when the point in question is at \mathbf { x } , and one associated with the body in its reference configuration, when the same point was at \mathbf { X } , because the orientation of the shell's reference surface--which determines our choice of basis vectors--will be quite different in these two configurations. We will write \mathbf { e } _ { \alpha } , \alpha = 1 , 2 , 3 , as the basis vectors chosen to write components associated with the current configuration