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Elements
\overline {{N}} = \rho N + (1 - \rho) \tilde {N}.
The tangent stiffness of the section behavior gives
\left\{ \begin{array}{l} d \tilde {N} \\ d M _ {1} \\ d M _ {2} \\ d M _ {3} \end{array} \right\} = \left[ \begin{array}{l l l l} A _ {0 0} & A _ {0 1} & A _ {0 2} & A _ {0 3} \\ & A _ {1 1} & A _ {1 2} & A _ {1 3} \\ \mathrm{sym} & & A _ {2 2} & A _ {2 3} \\ & & & A _ {3 3} \end{array} \right] \left\{ \begin{array}{l} d \varepsilon \\ d K _ {1} \\ d K _ {2} \\ d e _ {1} \end{array} \right\}.
If L ^ { 2 } A _ { 0 0 } < A _ { 1 1 } , A _ { 2 2 } (where L is the element length), then the beam is flexible axially and the mixed formulation is unnecessary. Otherwise, we assume that an inverse of the first equation above defines d" from d \tilde { N } :
d \varepsilon = \frac {1}{A _ {0 0}} (d \tilde {N} - A _ {0 1} d K _ {1} - A _ {0 2} d K _ {2} - A _ {0 3} d e _ {1}),
and so
d M _ {1} = \Big (A _ {1 1} - \frac {A _ {0 1} ^ {2}}{A _ {0 0}} \Big) d K _ {1} + \Big (A _ {1 2} - \frac {A _ {0 1} A _ {0 2}}{A _ {0 0}} \Big) d K _ {2} + \Big (A _ {1 3} - \frac {A _ {0 1} A _ {0 3}}{A _ {0 0}} \Big) d e _ {1} + \frac {A _ {0 1}}{A _ {0 0}} d \tilde {N}
d M _ {2} = \Big (A _ {1 2} - \frac {A _ {0 1} A _ {0 2}}{A _ {0 0}} \Big) d K _ {1} + \Big (A _ {2 2} - \frac {A _ {0 2} ^ {2}}{A _ {0 0}} \Big) d K _ {2} + \Big (A _ {2 3} - \frac {A _ {0 2} A _ {0 3}}{A _ {0 0}} \Big) d e _ {1} + \frac {A _ {0 2}}{A _ {0 0}} d \tilde {N}
d M _ {3} = \Big (A _ {1 3} - \frac {A _ {0 1} A _ {0 3}}{A _ {0 0}} \Big) d K _ {1} + \Big (A _ {2 3} - \frac {A _ {0 2} A _ {0 3}}{A _ {0 0}} \Big) d K _ {2} + \Big (A _ {3 3} - \frac {A _ {0 3} ^ {2}}{A _ {0 0}} \Big) d e _ {1} + \frac {A _ {0 3}}{A _ {0 0}} d \tilde {N}.
Now using the first tangent section stiffness multiplied by \rho and the second multiplied by 1 - \rho , the Newton contribution of the element becomes
\begin{array}{l} \int_ {L} \left\lfloor \delta \varepsilon \delta K _ {1} \delta K _ {2} \delta e _ {1} A _ {0 0} \delta \lambda \right\rfloor \left[ \tilde {A} \right] \left\{ \begin{array}{l} d \varepsilon \\ d K _ {1} \\ d K _ {2} \\ d e _ {1} \\ d \tilde {N} \end{array} \right\} d L \\ + \int_ {L} (\tilde {N} d \delta \varepsilon + M _ {1} d \delta K _ {1} + M _ {2} d \delta K _ {2} + M _ {3} d \delta e _ {1}) d L \\ = - \int_ {L} \left[ \tilde {N} \delta \varepsilon + M _ {1} \delta K _ {1} + M _ {2} \delta K _ {2} + M _ {3} \delta e _ {1} + A _ {0 0} \delta \lambda (1 - \rho) \left(\frac {N - \tilde {N}}{A _ {0 0}}\right) \right] d L, \\ \end{array}
where £ A~ ¤ is
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\left[ \begin{array}{cccc} \rho A _ {0 0} & \rho A _ {0 1} & \rho A _ {0 2} & \rho A _ {0 3} & 1 - \rho \\ & A _ {1 1} - (1 - \rho) \frac {A _ {0 1} ^ {2}}{A _ {0 0}} & A _ {1 2} - (1 - \rho) \frac {A _ {0 1} A _ {0 2}}{A _ {0 0}} & A _ {1 3} - (1 - \rho) \frac {A _ {0 1} A _ {0 3}}{A _ {0 0}} & (1 - \rho) \frac {A _ {0 1}}{A _ {0 0}} \\ & & A _ {2 2} - (1 - \rho) \frac {A _ {0 2} ^ {2}}{A _ {0 0}} & A _ {2 3} - (1 - \rho) \frac {A _ {0 2} A _ {0 3}}{A _ {0 0}} & (1 - \rho) \frac {A _ {0 2}}{A _ {0 0}} \\ \mathrm{symm} & & & A _ {3 3} - (1 - \rho) \frac {A _ {0 3} ^ {2}}{A _ {0 0}} & (1 - \rho) \frac {A _ {0 3}}{A _ {0 0}} \\ & & & & - (1 - \rho) \frac {1}{A _ {0 0}} \\ \end{array} \right].
The variable \tilde { N } is taken as an independent value at each integration point in the element. We choose \rho as \tilde { \rho } / A _ { 0 0 } , where \tilde { \rho } is a small value. With this choice, by ensuring that the variables \tilde { N } are eliminated after the displacement variables of each element, the Gaussian elimination scheme has no difficulty with solving the equations.
Transverse shear
In the mixed elements that allow transverse shear (B21H, B22H, B31H, B32H), the transverse shear constraints are imposed by treating the shear forces as independent variables, using the following formulation. The internal virtual work associated with transverse shear is
\delta W _ {1} ^ {T S} = \int_ {L} T _ {i} \delta \gamma_ {i} d L \qquad i = 1, 2,
where T _ { 1 } and T _ { 2 } are shear forces on the section, and \delta \gamma _ { 1 } and \delta \gamma _ { 2 } are variations of transverse shear strain. The virtual work can also be written by introducing independent shear force variables \tilde { T } _ { 1 } and \tilde { T } _ { 2 } . , as
\delta W _ {2} ^ {T S} = \int \{\tilde {T} _ {i} \delta \gamma_ {i} + \delta \lambda_ {i} (T _ {i} - \tilde {T} _ {i}) \} d L,
where the \delta \lambda _ { i } are Lagrange multipliers. As in the axial case, we take a linear combination of these two forms,
\delta W ^ {T S} = \rho \delta W _ {1} ^ {T S} + (1 - \rho) \delta W _ {2} ^ {T S},
where \rho will be defined later. This gives
\delta W ^ {T S} = \int_ {L} \left\{\overline {{{T}}} _ {i} \delta \gamma_ {i} + \delta \lambda_ {i} (1 - \rho) (T _ {i} - \tilde {T} _ {i}) \right\} d L,
where
\overline {{T}} _ {i} = \rho T _ {i} + (1 - \rho) \tilde {T} _ {i}.
The contribution of this term to the Newton scheme is
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\begin{array}{l} \int_ {L} \left\{(\rho d T _ {i} + (1 - \rho) d \tilde {T} _ {i}) \delta \gamma_ {i} + \delta \lambda_ {i} (1 - \rho) (d T _ {i} - d \tilde {T} _ {i}) + \tilde {T} _ {i} d \delta \gamma_ {i} \right\} d L \\ = - \int_ {L} \left\{\overline {{{T}}} _ {i} \delta \gamma_ {i} + \delta \lambda_ {i} (1 - \rho) (T _ {i} - \tilde {T} _ {i}) \right\} d L. \\ \end{array}
ABAQUS treats transverse shear elastically, so T _ { i } = G A \gamma _ { i } , where GA is constant. Then the Newton contribution is
\begin{array}{l} \int_ {L} \left\{(\rho G A d \gamma_ {i} + (1 - \rho) d \tilde {T} _ {i}) \delta \gamma_ {i} + \delta \lambda_ {1} (1 - \rho) (G A d \gamma_ {i} - d \tilde {T} _ {i}) + \overline {{{T}}} _ {i} d \delta \gamma_ {i} \right\} d L \\ = - \int_ {L} \left\{\overline {{{T}}} _ {i} \delta \gamma_ {i} + \delta \lambda_ {i} (1 - \rho) (T _ {i} - \overline {{{T}}} _ {i}) \right\} d L. \\ \end{array}
We now define \delta \tilde { T } _ { i } = G A \delta \lambda _ { i } and choose \rho = \tilde { \rho } / G A , where \tilde { \rho } is a small value compared to G A , , to give
\begin{array}{l} \int_ {L} \left\{\tilde {\rho} d \gamma_ {i} + (1 - \tilde {\rho} / G A) d \tilde {T} _ {i}) \delta \gamma_ {i} + (1 - \tilde {\rho} / G A) (d \gamma_ {i} - \frac {1}{G A} d \tilde {T} _ {i}) \delta T _ {i} + \overline {{T}} _ {i} d \delta \gamma_ {i} \right\} d L \\ = - \int_ {L} \left\{\overline {{{T}}} _ {i} \delta \gamma_ {i} + (1 - \tilde {\rho} / G A) (\gamma_ {i} - \tilde {T} _ {i} / G A) \delta \tilde {T} _ {i} \right\} d L. \\ \end{array}
3.5.5 Pressure load stiffness for beam elements
ABAQUS provides for loads per unit length in the beam cross-sectional directions as distributed load options for the beam elements (load types P1, P2). Since these are follower forces, they have a load stiffness; and this stiffness can sometimes be important especially in the case of buckling prediction by eigenvalue extraction. The symmetric form of this load stiffness is included in ABAQUS/Standard (see Hibbitt, 1979, and Mang, 1980). This form is developed below. The external virtual work on the beam is
\delta \mathbf {W} ^ {e} = \int_ {S} \mathbf {p} \cdot \delta \mathbf {u} d S,
where the pressure load, p, is given by the externally prescribed pressure magnitude, p , \mathbf { a s } \ \mathbf { p } = p \mathbf { n } _ { \alpha } where ® = 1 or 2 defines the particular cross-sectional direction of the load. Therefore,
{ \bf n } _ { \alpha } ~ = ~ ( - 1 ) ^ { \beta } { \bf n } _ { \beta } \times ( d { \bf x } / d S ) , where \beta = 2 when ® = 1, and \beta = 1 when \alpha = 2 so that
Elements
\delta \mathbf {W} ^ {e} = (- 1) ^ {\beta} \int_ {S} p \left(\mathbf {n} _ {\beta} \times \frac {d \mathbf {x}}{d S}\right) \cdot \delta \mathbf {u} d S,
where S is the material coordinate along the beam. Now assuming that the load magnitude, p, is externally prescribed so that it does not change with position, the rate of change of ±We with change in position, du, is
d \delta \mathbf {W} ^ {e} = (- 1) ^ {\beta} \int_ {S} p \left(d \mathbf {n} _ {\beta} \times \frac {d \mathbf {x}}{d S} + \mathbf {n} _ {\beta} \times \frac {d d \mathbf {u}}{d S}\right) \cdot \delta \mathbf {u} d S.
Now
d \mathbf {n} _ {\beta} = d \boldsymbol {\omega} \times \mathbf {n} _ {\beta},
and so
d \mathbf {n} _ {\beta} \times \frac {d \mathbf {x}}{d S} = (d \boldsymbol {\omega} \times \mathbf {n} _ {\beta}) \cdot \frac {d \mathbf {x}}{d S} = (d \boldsymbol {\omega} \cdot \frac {d \mathbf {x}}{d S}) \mathbf {n} _ {\beta}, \mathrm{neglecting} \mathbf {n} _ {\beta} \cdot \frac {d \mathbf {x}}{d S}.
Thus,
d \delta \mathbf {W} ^ {e} = (- 1) ^ {\beta} \int_ {S} p \left[ d \pmb {\omega} \cdot \frac {d \mathbf {x}}{d S} \mathbf {n} _ {\beta} \cdot \delta \mathbf {u} + \left(\mathbf {n} _ {\beta} \times \frac {d d \mathbf {u}}{\mathrm{d} S}\right) \cdot \delta \mathbf {u} \right] d S.
This load stiffness is not symmetric, except in the case of a beam in a plane with fixed ends (or no ends, such as a ring), in which case the first term is exactly zero and the second gives the symmetric form
(- 1) ^ {\beta} \int_ {S} \frac {1}{2} p \mathbf {n} _ {\beta} \cdot \left(\frac {d d \mathbf {u}}{\mathrm{d} S} \times \delta \mathbf {u} + \frac {d \mathbf {u}}{d S} \times d \delta \mathbf {u}\right) d S.
In ABAQUS, even for the general beams in three dimensions, the load stiffness is introduced as the symmetric part of d±We above.
3.6 Shell elements
3.6.1 Shell element overview
The ABAQUS shell element library provides elements that allow the modeling of curved, intersecting shells that can exhibit nonlinear material response and undergo large overall motions (translations and rotations). ABAQUS shell elements can also model the bending behavior of composites.
The library is divided into three categories consisting of general-purpose, thin, and thick shell elements. Thin shell elements provide solutions to shell problems that are adequately described by
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classical (Kirchhoff) shell theory, thick shell elements yield solutions for structures that are best modeled by shear flexible (Mindlin) shell theory, and general-purpose shell elements can provide solutions to both thin and thick shell problems. All shell elements use bending strain measures that are approximations to those of Koiter-Sanders shell theory ( Budiansky and Sanders, 1963). While ABAQUS/Standard provides shell elements in all three categories, ABAQUS/Explicit provides only general-purpose shell elements. For most applications the general-purpose shell elements should be the user's first choice from the element library. However, for specific applications it may be possible to obtain enhanced performance by choosing one of the thin or thick shell elements. It should also be noted that not all ABAQUS shell elements are formulated for large-strain analysis.
The general-purpose shell elements are axisymmetric elements SAX1, SAX2, and SAX2T and three-dimensional elements S3, S4, S3R, S4R, S4RS, S3RS, and S4RSW, where S4RS, S3RS, and S4RSW are small-strain elements that are available only in ABAQUS/Explicit. The general-purpose elements provide robust and accurate solutions in all loading conditions for thin and thick shell problems. Thickness change as a function of in-plane deformation is allowed in their formulation. They do not suffer from transverse shear locking, nor do they have any unconstrained hourglass modes. With the exception of the small-strain elements, all of these elements consider finite membrane strains. No hourglass control is required for the axisymmetric general-purpose shells, nor in the bending and membrane response of the fully integrated element S4. The membrane kinematics of S4 are based on an assumed-strain formulation that provides accurate solutions for in-plane bending behavior. The ABAQUS/Explicit elements S3RS, S4RS, and S4RSW are well-suited for many impact dynamics problems, including structures undergoing large-scale buckling behavior, which involve small-strains but large rotations and severe bending. These elements use simplified methods for strain calculation and hourglass control and offer significant advantages in computational speed.
Thin shell elements are available only in ABAQUS/Standard. STRI3 and STRI65 are triangular small-strain, thin shell elements; S4R5, S8R5, and S9R5 comprise the quadrilateral small-strain, thin shell elements, while SAXA is a finite-strain, thin shell element suitable for modeling axisymmetric geometries subjected to arbitrary loadings. Thin shell elements may provide enhanced performance for large problems where reducing the number of degrees of freedom through the use of five degree of freedom shells is desirable. However, they should be used only for the modeling of thin structures that exhibit at most weak nonlinearities in problems where rotation degree of freedom output is notexhibit at most weak nonlinearities in problems where rotation degree of freedom output is not required and for situations where the shell surface and the displacement field are smooth so that higher accuracy can be achieved with the use of second-order shells. SAXA elements very effectively model axisymmetric structures undergoing asymmetric deformation when only a few circumferential Fourieraxisymmetric structures undergoing asymmetric deformation when only a few circumferential Fourier modes describe the circumferential variation of the deformation accurately.
The Discrete Kirchhoff (DK) constraint, which refers to the satisfaction of the Kirchhoff constraint at discrete points on the shell surface, is imposed in all thin shell elements in ABAQUS. For element type STRI3 the constraint is imposed analytically and involves no transverse shear strain energy calculation. Solutions obtained with these elements converge to those corresponding to classical shell theory. For element types STRI65, S4R5, S8R5, S9R5, and SAXA the discrete Kirchhoff constraint is imposed numerically where the transverse shear stiffness acts as a penalty that enforces the constraint.
Shell behavior that can be properly described with shear flexible shell theory and results in smooth displacement fields can be analyzed accurately with the second-order ABAQUS/Standard thick shell
element S8R. Nonnegligible transverse shear flexibility is required for this element to function properly; hence, the element is suitable for the analysis of composite and sandwich shells. Irregular meshes of S8R elements converge very poorly because of severe transverse shear locking; therefore,of severe this element is recommended for use in regular mesh geometries for thick shell applications.
Thickness change
In geometrically nonlinear analyses in ABAQUS/Standard the cross-section thickness of finite-strain shell elements changes as a function of the membrane strain based on a user-defined "effective section Poisson's ratio," º. In ABAQUS/Explicit the thickness change is based on the "effective section Poisson's ratio" for all shell elements in large-deformation analyses where the POISSON parameter is not set to MATERIAL. The thickness change based on the "effective section Poisson's ratio" is calculated as follows.
In plane stress \sigma _ { 3 3 } = 0 ; linear elasticity gives
\epsilon_ {3 3} = - \frac {\nu}{1 - \nu} (\epsilon_ {1 1} + \epsilon_ {2 2}).
Treating these as logarithmic strains,
\ln \left(\frac {t}{t _ {0}}\right) = - \frac {\nu}{1 - \nu} \left(\ln \left(\frac {l _ {1}}{l _ {1} ^ {0}}\right) + \ln \left(\frac {l _ {2}}{l _ {2} ^ {0}}\right)\right) = - \frac {\nu}{1 - \nu} \ln \left(\frac {A}{A ^ {0}}\right),
where A is the area on the shell's reference surface. This nonlinear analogy with linear elasticity leads to the thickness change relationship:
{\frac {t}{t _ {0}}} = \left({\frac {A}{A _ {0}}}\right) ^ {- {\frac {\nu}{1 - \nu}}}.
For \nu = 0 . 5 the material is incompressible; for \nu = 0 . 0 the section thickness does not change.
3.6.2 Axisymmetric shell elements
These two shell elements are axisymmetric versions of the shells described in the previous section and use the "reduced-integration penalty" method of Hughes et al. (1977). While these are shell elements, they are also simple extensions of the two-dimensional beam elements B21 and B22. The extension is the inclusion of the hoop terms. These elements are thus one-dimensional, deforming in a radial plane. The Cartesian coordinates in this plane are r (radius) and z (axial position). Distance along the shell reference surface in such a plane is measured by the material coordinate S (see Figure 3.6.2-1).
Figure 3.6.2-1 Axisymmetric shell.
text_image
z S Shell middle surface n Shell normal r
Interpolation and integration
The 2-node element (SAX1) uses one-point integration of the linear interpolation function for the distribution of loads. The mass matrix is lumped. The 3-node element ( SAX2) uses two-point integration of a quadratic interpolation function for the stiffness and three-point integration of a quadratic interpolation function for the distribution of loads. SAX2 uses a consistent mass matrix. All integrations use the Gauss method.
Theory
This shell theory allows for finite strains and rotations of the shell. The strain measure used is chosen to give a close approximation (accurate to second-order terms) to log strain. Thus, the theory is intended for direct application to cases involving inelastic or hypoelastic deformation where the stress-strain behavior is given in terms of Kirchhoff stress ("true" stress in the usual engineering literature) and log strain, such as metal plasticity. The theory is approximate, but the approximations are not rigorously justified: they are introduced for simplicity and seem reasonable. These approximations are as follows:
a. A "thinness" assumption is made. This means that, at all times, only terms up to first order with respect to the thickness direction coordinate are included.
b. The thinning of the shell caused by stretching parallel to its reference surface is assumed to be uniform through the thickness and defined by an incompressibility condition on the reference surface of the shell. Obviously this is a relatively coarse approximation, especially in the case where a shell is subjected to pure bending. It is adopted because it is simple and models the effect of thinning associated with membrane straining: this is considered to be of primary importance in the type of applications envisioned, such as the failure of pipes and vessels subjected to over-pressurization.
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c. The thinning of the shell is assumed to occur smoothly--that is to say, gradients of the thinning with respect to position on the reference surface are assumed to be negligible. This means that localization effects, such as necking of the shell, are only modeled in a very coarse way. Again, the reason for adopting this approximation is simplicity--details of localization effects are not important to the type of application for which the elements are designed.
d. All stresses except those parallel to the reference surface are neglected; and, for the nonnegligible stresses, plane stress theory is assumed. As with (c) above, this precludes detailed localization studies, but introduces considerable simplification into the formulation.
e. Plane sections remain plane. This has been shown to be consistent with the thinness assumption, (a) above, for most material models. Here it is simply assumed without further justification.
f. Transverse shears are assumed to be small, and the material response to such deformation is assumed to be linear elastic. Transverse shear is introduced because the elements used are of the "reduced integration, penalty" type (see Hughes et al., 1977, for example). In these elements position on the reference surface and rotation of lines initially orthogonal to the reference surface are interpolated independently: the transverse shear stiffness is then viewed as a penalty term imposing the necessary constraint at selected (reduced integration) points. This transverse shear stiffness is the actual elastic value for relatively thick shells. For thinner cases the penalty must be reduced for numerical reasons--this is done in ABAQUS in the manner described in Hughes et al. (1977).
The theory is now described in detail. The concepts are taken from various sources, most especially Budiansky and Sanders (1963) and Rodal and Witmer (1979). The position of a material point in the shell is given by
Equation 3.6.2-1
\mathbf {x} ^ {1} = \mathbf {x} + \lambda_ {t} \eta \mathbf {n},
where
\mathbf {x} (\theta^ {1}, \theta^ {2})
is the position of a point on the reference surface of the shell;
\mathbf {n} (\theta^ {1}, \theta^ {2})
is a unit vector in the "thickness" direction, this direction being initially orthogonal to the reference surface;
\lambda_ {t}
is the stretch of the shell in the thickness direction;
\eta
measures position with respect to the thickness direction, in the reference configuration; and
(\theta^ {1}, \theta^ {2})
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are material coordinates in the reference surface.
The assumptions listed above imply that \lambda _ { t } = \lambda _ { t } ( \theta ^ { 1 } , \theta ^ { 2 } ) only and that \partial \lambda _ { t } / \partial \theta ^ { 1 } , \ \partial \lambda _ { t } / \partial \theta ^ { 2 } are small quantities. Equation 3.6.2-1 is written at the end of an increment, and at the start of an increment the same equation is written as
Equation 3.6.2-2
\mathbf {X} ^ {1} = \mathbf {X} + \lambda_ {t} ^ {o} \eta \mathbf {N}.
The metric at the end of an increment is
Equation 3.6.2-3
\begin{array}{l} \frac {\partial \mathbf {x} ^ {1}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {x} ^ {1}}{\partial \theta^ {\beta}} = \left(\frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} + \lambda_ {t} \eta \frac {\partial \mathbf {n}}{\partial \theta^ {\alpha}}\right) \cdot \left(\frac {\partial \mathbf {x}}{\partial \theta^ {\beta}} + \lambda_ {t} n \frac {\partial \mathbf {n}}{\partial \theta^ {\beta}}\right) \\ \approx \frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial \theta^ {\beta}} + \eta \lambda_ {t} \left(\frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {n}}{\partial \theta^ {\beta}} + \frac {\partial \mathbf {n}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial \theta^ {\beta}}\right) \\ = g _ {\alpha \beta} + \eta \lambda_ {t} b _ {\alpha \beta}, \mathrm{say}, \\ \end{array}
where
g _ { \alpha \beta } = \frac { \partial \mathbf { x } } { \partial \theta ^ { \alpha } } \cdot \frac { \partial \mathbf { x } } { \partial \theta ^ { \beta } } @x @x is the metric of the reference surface,
and
b _ {\alpha \beta} = \frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {n}}{\partial \theta^ {\beta}} + \frac {\partial \mathbf {n}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {x}}{\partial \theta^ {\beta}}
is an approximation to the curvature tensor (second fundamental form) of the reference surface. b _ { \alpha \beta } would be precisely the curvature tensor as it is usually defined if
\mathbf {n} \cdot \frac {\partial \mathbf {x}}{\partial \theta^ {\alpha}} = 0.
This is only approximately true for these elements, because a small transverse shear is allowed.
At the start of the increment the same quantities are
Equation 3.6.2-4
\frac {\partial \mathbf {X} ^ {1}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {X} ^ {1}}{\partial \theta^ {\beta}} = G _ {\alpha \beta} + \eta \lambda_ {t} ^ {o} B _ {\alpha \beta}.
Axisymmetric shells undergoing axisymmetric deformations have the great simplification that principal directions do not rotate. Thus, by assuming that \theta ^ { 1 } and \theta ^ { 2 } are oriented in these principal directions ( \theta ^ { 1 } is meridional and \theta ^ { 2 } is circumferential), the stretch ratios that occur within the
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increment in these directions are written as
\Delta \lambda_ {\alpha} ^ {1} = \left(\frac {\partial \mathbf {x} ^ {1}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {x} ^ {1}}{\partial \theta^ {\alpha}} \right. / \left. \frac {\partial \mathbf {X} ^ {1}}{\partial \theta^ {\alpha}} \cdot \frac {\partial \mathbf {X} ^ {1}}{\partial \theta^ {\alpha}}\right) ^ {- \frac {1}{2}},
where from this point onward the summation convention has been dropped. Using Equation 3.6.2-3 and Equation 3.6.2-4 and truncating to first order in ´ then gives
Equation 3.6.2-5
\Delta \lambda_ {\alpha} ^ {1} \approx \Delta \lambda_ {\alpha} (1 + \eta \Delta k _ {\alpha \alpha}),
where
\Delta \lambda_ {\alpha} = \left(g _ {\alpha \alpha} \mathord {\left/ {g _ {\alpha \alpha} \right.} \left. \right.} G _ {\alpha \alpha}) ^ {\frac {1}{2}}
and
\Delta k _ {\alpha \alpha} = \lambda_ {t} \frac {b _ {\alpha \alpha}}{g _ {\alpha \alpha}} - \lambda_ {t} ^ {o} \frac {B _ {\alpha \alpha}}{G _ {\alpha \alpha}}.
The incremental strain, \Delta \varepsilon _ { \alpha \alpha } ^ { 1 } , is defined as
\Delta \varepsilon_ {\alpha \alpha} ^ {1} = 2 \left(\frac {\Delta \lambda_ {\alpha} ^ {1} - 1}{\Delta \lambda_ {\alpha} ^ {1} + 1}\right).
Because this expression approximates the increment of log strain correctly to second-order terms, it can be thought of as a central difference approximation for the rate of deformation. This expression is used because we anticipate that strain increments of a maximum of 20 percent per increment will be used: at that magnitude the difference between this definition of incremental strain and the increment of log strain is about 1%, which seems to be acceptable (4 % of the increment). At lower--and probably more typical--values of strain increment, the error is very much less. Again expanding to first order in the thickness direction coordinate, ´, we obtain
\Delta \varepsilon_ {\alpha \alpha} ^ {1} \approx \Delta \varepsilon_ {\alpha \alpha} + \eta \frac {4 \Delta \lambda_ {\alpha}}{(1 + \Delta \lambda_ {\alpha}) ^ {2}} \Delta k _ {\alpha \alpha},
where \Delta \varepsilon _ { \alpha \alpha } is the incremental strain of the reference surface--the membrane strain. Now consider the term
\frac {4 \Delta \lambda_ {\alpha}}{(1 + \Delta \lambda_ {\alpha}) ^ {2}}.
Write \Delta \lambda _ { \alpha } = 1 + e , where e represents the change in length per unit length that occurs within the increment (the "nominal strain" with respect to the configuration at the beginning of the increment).