김경종
32 KiB
\frac {\partial^ {t} x _ {2}}{\partial s} = \frac {h}{2} \left[ \left(\frac {1 - r}{2}\right) \cos^ {t} \theta_ {1} + \left(\frac {1 + r}{2}\right) \cos^ {t} \theta_ {2} \right]
where we assumed ^{1}L = ^{0}L = L .
Next we consider the TL formulation. Here we use
{ } ^ { 0 } \mathbf { J } = \left[ \begin{array} { c c c } \frac { 0 L } { 2 } & & 0 \\ 0 & & \frac { h } { 2 } \end{array} \right]
Also, the initial displacement effect is taken into account using the derivatives
_ 0 u _ {1, 1} = (\cos \alpha - 1) - \frac {s h}{2 L} (\sin^ {\prime} \theta_ {2} - \sin^ {\prime} \theta_ {1})
{ } _ { 0 } ^ { \prime } u _ { 1 , 2 } = - \left( \frac { 1 - r } { 2 } \right) \sin { { } ^ { \prime } \theta _ { 1 } } - \left( \frac { 1 + r } { 2 } \right) \sin { { } ^ { \prime } \theta _ { 2 } }
_ 0 u _ {2, 1} = \sin \alpha + \frac {s h}{2 L} \left(\cos^ {\prime} \theta_ {2} - \cos^ {\prime} \theta_ {1}\right)
_ {0} u _ {2, 2} = \left(\frac {1 - r}{2}\right) \cos^ {\prime} \theta_ {1} + \left(\frac {1 + r}{2}\right) \cos^ {\prime} \theta_ {2} - 1
where we again assumed ^{1}L = ^{0}L = L .
In each case, we note that these expressions lead to the strain terms corresponding to the global stationary coordinate system. These terms must be transformed to the local \eta , \xi axes for construction of the strain-displacement matrix of the element.
Finally, we should note that the element can be employed in plane stress or plane strain conditions, depending on the stress-strain relation used (see Section 4.2.3). In plane stress analysis the thickness of the element (normal to the x_{1} , x_{2} plane) must of course be given (this thickness is assumed to be unity in plane strain analysis).
EXAMPLE 6.21: The two-node element in Example 6.20 is to be used as a shell element in axisymmetric conditions. Discuss what terms in addition to those given in Example 6.20 need to be included in the construction of the strain-displacement matrices for the TL formulation.
In axisymmetric analysis the integration is performed over 1 radian and the hoop strain effect must be included (see Example 5.9). Table 6.5 gives the incremental hoop strain _{0}\epsilon_{33} , which must be evaluated using the interpolations stated in Example 6.20 to give a third row in the strain-displacement matrices _{0}^{t}B_{L0} and _{0}^{t}B_{L1} . The third row of the matrix _{0}^{t}B_{L0} corresponds to the term u_{1}/^{0}x_{1} , hence,
{ } _ { 0 } ^ { t } \mathbf { B } _ { L 0 } = \left[ \begin{array} { c c c c c c c c c } \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ \frac { 1 - r } { 2 ^ { 0 } x _ { 1 } } & 0 & \frac { - s h } { 2 } \left( \frac { 1 - r } { 2 } \right) \frac { \cos ^ { \prime } \theta _ { 1 } } { ^ { 0 } x _ { 1 } } & \frac { 1 + r } { 2 ^ { 0 } x _ { 1 } } & 0 & \frac { - s h } { 2 } \left( \frac { 1 + r } { 2 } \right) \frac { \cos ^ { \prime } \theta _ { 2 } } { ^ { 0 } x _ { 1 } } \end{array} \right]
where we have used the following ordering of nodal variables in the solution vector
\hat {\mathbf {u}} ^ {T} = \left[ u _ {1} ^ {1} \quad u _ {2} ^ {1} \quad \theta_ {1} \quad u _ {1} ^ {2} \quad u _ {2} ^ {2} \quad \theta_ {2} \right]
and ^0 x_1 = [(1 + r)/2] L . The third row of the matrix ^0 \mathbf{B}_{L1} corresponds to the strain term ^t u_1 u_1 / (^0 x_1)^2 and for its evaluation the interpolations of ^t u_1 , ^0 x_1 , and u_1 are similarly used.
The terms in the nonlinear strain stiffness matrix corresponding to _{0}S_{33} are evaluated from the expression
\begin{array}{l} { } _ { 0 } ^ { \prime } S _ { 3 3 } \left\{ \delta \theta _ { 1 } \left[ \frac { s h } { 2 } \left( \frac { 1 - r } { 2 } \right) \frac { \sin { } ^ { \prime } \theta _ { 1 } } { ^ { 0 } x _ { 1 } } \left( 1 + \frac { { } ^ { \prime } u _ { 1 } } { ^ { 0 } x _ { 1 } } \right) \right] \theta _ { 1 } + \delta \theta _ { 2 } \left[ \frac { s h } { 2 } \left( \frac { 1 + r } { 2 } \right) \frac { \sin { } ^ { \prime } \theta _ { 2 } } { ^ { 0 } x _ { 1 } } \left( 1 + \frac { { } ^ { \prime } u _ { 1 } } { ^ { 0 } x _ { 1 } } \right) \right] \theta _ { 2 } \right. \\ + \left(\frac {1 - r}{2 ^ {0} x _ {1}} \delta u _ {1} ^ {1} - \left[ \frac {s h}{2} \left(\frac {1 - r}{2}\right) \frac {\cos^ {\prime} \theta_ {1}}{^ {0} x _ {1}} \right] \delta \theta_ {1} + \frac {1 + r}{2 ^ {0} x _ {1}} \delta u _ {1} ^ {2} - \left[ \frac {s h}{2} \left(\frac {1 + r}{2}\right) \frac {\cos^ {\prime} \theta_ {2}}{^ {0} x _ {1}} \right] \delta \theta_ {2}\right) \\ \left. \times \left(\frac {1 - r}{2 ^ {0} x _ {1}} u _ {1} ^ {1} - \left[ \frac {s h}{2} \left(\frac {1 - r}{2}\right) \frac {\cos^ {\prime} \theta_ {1}}{^ {0} x _ {1}} \right] \theta_ {1} + \frac {1 + r}{2 ^ {0} x _ {1}} u _ {1} ^ {2} - \left[ \frac {s h}{2} \left(\frac {1 + r}{2}\right) \frac {\cos^ {\prime} \theta_ {2}}{^ {0} x _ {1}} \right] \theta_ {2}\right) \right\} \\ \end{array}
This expression is of the form \delta \hat{\mathbf{u}}^T (\mathbf{\delta}_{0}\mathbf{K}_{NL}^{*})\hat{\mathbf{u}} , where \mathbf{K}_{NL}^{*} represents a contribution to the element nonlinear strain stiffness matrix.
6.5.2 Plate and General Shell Elements
Many plate and shell elements have been proposed for the nonlinear analysis of plates, specific shells, and general shell structures. However, as with the beam element discussed in the previous section, the isoparametric formulations of plate and shell elements for nonlinear analysis are very attractive because these formulations are both consistent and general, and the elements can be employed in an effective manner for the analysis of a variety of plates and shells. As in linear analysis, in essence a very general shell theory is employed in the formulation so that the shell elements are applicable, in principle, to the analysis of any plate and shell structure.
Considering a plate undergoing large deflections, we recognize that as soon as the plate has deflected significantly, the action of the structure is really that of a shell; i.e., the structure is now curved, and both membrane and bending stresses are significant. Therefore, in the discussion below we consider only general shell elements, where we imply that if a specific element is initially flat, it represents a plate.
In the following presentation we consider the nonlinear formulation of the MITC shell elements discussed for linear analysis in Section 5.4.2. Figure 6.6 shows a typical nine-node
text_image
At time t At time zero x₃, t₃ e₃ e₂ e₁ x₂, t₂ x₁, t₁ t₃Vₙ³ t₃V₂³ t₃V₁³ t₃Vₙ³ t₃V₂³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t₃Vₙ³ t s r s t s r s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s t s 1 2 3 4 5 6 7 8 9 At time t At time zero
Figure 6.6 Shell element undergoing large displacements and rotations
element in its original position and its configuration at time t. The element behavior is based on the same assumptions that are employed in linear analysis, namely, that straight lines defined by the nodal director vectors (which, usually, give lines that in the original configuration are close to normal to the midsurface of the shell) remain straight during the element deformations and that no transverse normal stress is developed in the directions of the director vectors. However, the nonlinear formulation given here does admit arbitrarily large displacements and rotations of the shell element. ^{10}
The UL and TL formulations of the shell element are based on the general continuum mechanics equations presented in Section 6.2.3 and are a direct extension of the formulation for linear analysis. Also, the calculation of the element matrices follows closely the calculations used for the beam elements (see Section 6.5.1).
Using the same notation as in Section 5.4.2, the coordinates of a generic point in the shell element now undergoing very large displacements and rotations are (see K. J. Bathe and S. Bolourchi [B])
^ \prime x _ {i} = \sum_ {k = 1} ^ {q} h _ {k} ^ {\prime} x _ {i} ^ {k} + \frac {t}{2} \sum_ {k = 1} ^ {q} a _ {k} h _ {k} ^ {\prime} V _ {n i} ^ {k} \tag {6.167}
Using (6.167) at times 0, t, and t + \Delta t , we thus have
{ } ^ { t } u _ { i } = { } ^ { t } x _ { i } - { } ^ { 0 } x _ { i } \tag {6.168}
and u_{i} = ^{t + \Delta t}x_{i} - ^{t}x_{i} (6.169)
Substituting from (6.167) into (6.168) and (6.169), we obtain
{ } ^ { \prime } u _ { i } = \sum _ { k = 1 } ^ { q } h _ { k } { } ^ { \prime } u _ { i } ^ { k } + \frac { t } { 2 } \sum _ { k = 1 } ^ { q } a _ { k } h _ { k } \left( { } ^ { \prime } V _ { n i } ^ { k } - { } ^ { 0 } V _ { n i } ^ { k } \right) \tag {6.170}
and u_{i} = \sum_{k = 1}^{q}h_{k}u_{i}^{k} + \frac{t}{2}\sum_{k = 1}^{q}a_{k}h_{k}V_{ni}^{k} (6.171)
where V_{ni}^{k} = ^{t+\Delta t}V_{ni}^{k} - ^{t}V_{ni}^{k} (6.172)
The relation in (6.170) is employed to evaluate the total displacements and total strains (hence also total stresses for both the UL and TL formulations) of the particles in the element. To apply (6.171) the same thoughts as in the beam element formulation for use of (6.158), (6.161), and (6.162) are applicable. Now we express the vector components V_{ni}^{k} in terms of rotations about two vectors that are orthogonal to V_{n}^{k} . These two vectors V_{1}^{k} and V_{2}^{k} are defined at time 0 (as in linear analysis) using
{ } ^ { 0 } \mathbf { V } _ { 1 } ^ { k } = \frac { \mathbf { e } _ { 2 } \times { } ^ { 0 } \mathbf { V } _ { n } ^ { k } } { \| \mathbf { e } _ { 2 } \times { } ^ { 0 } \mathbf { V } _ { n } ^ { k } \| _ { 2 } } \tag {6.173}
{ } ^ { 0 } \mathbf { V } _ { 2 } ^ { k } = { } ^ { 0 } \mathbf { V } _ { n } ^ { k } \times { } ^ { 0 } \mathbf { V } _ { 1 } ^ { k } \tag {6.174}
where we set ^{0}V_{1}^{k} equal to e_{3} if ^{0}V_{n}^{k} is parallel to e_{2} . The vectors for time t are then obtained by an integration process briefly described for the director vector in (6.177).
Let \alpha_{k} and \beta_{k} be the rotations of the director vector ^t\mathbf{V}_n^k about the vectors ^t\mathbf{V}_1^k and ^t\mathbf{V}_2^k in the configuration at time t . Then we have approximately for small angles \alpha_{k} and \beta_{k} , but including second-order rotation effects (see Exercise 6.57),
\mathbf {V} _ {n} ^ {k} = - ^ {t} \mathbf {V} _ {2} ^ {k} \alpha_ {k} + ^ {t} \mathbf {V} _ {1} ^ {k} \beta_ {k} - \frac {1}{2} \left(\alpha_ {k} ^ {2} + \beta_ {k} ^ {2}\right) ^ {t} \mathbf {V} _ {n} ^ {k} \tag {6.175}
We include the quadratic terms in rotations because we want to arrive at the consistent tangent stiffness matrix, and these terms contribute to the nonlinear strain stiffness effects. Namely, substituting from (6.175) into (6.171), we obtain
u _ {i} = \sum_ {k = 1} ^ {q} h _ {k} u _ {i} ^ {k} + \frac {t}{2} \sum_ {k = 1} ^ {q} a _ {k} h _ {k} \left[ - ^ {\prime} V _ {2 i} ^ {k} \alpha_ {k} + ^ {\prime} V _ {1 i} ^ {k} \beta_ {k} - \frac {1}{2} \left(\alpha_ {k} ^ {2} + \beta_ {k} ^ {2}\right) ^ {\prime} V _ {n i} ^ {k} \right] \tag {6.176}
Using this expression to evaluate the continuum terms in Tables 6.2 and 6.3, we notice that the terms \int_{t_V}{}^t\tau_{ij}\delta_t e_{ij}d^t V and \int_{0_V}{}^t S_{ij}\delta_0e_{ij}d^0 V result in a stiffness contribution due to the quadratic terms in (6.176) that we naturally add to the other terms of the nonlinear strain stiffness matrix.
We arrived at a similar result in the formulation of the isoparametric beam elements discussed in the previous section [see (6.161) and (6.162) and the ensuing discussion].
The finite element solution will yield the nodal point variables u_{i}^{k} , \alpha_{k} , and \beta_{k} , which can then be employed to evaluate ^{t + \Delta t}\mathbf{V}_n^k ,
{ } ^ { t + \Delta t } \mathbf { V } _ { n } ^ { k } = { } ^ { t } \mathbf { V } _ { n } ^ { k } + \int _ { \alpha _ { k } , \beta _ { k } } - { } ^ { \tau } \mathbf { V } _ { 2 } ^ { k } d \alpha _ { k } + { } ^ { \tau } \mathbf { V } _ { 1 } ^ { k } d \beta _ { k } \tag {6.177}
This integration can be performed in one step using an orthogonal matrix for finite rotations (see, for example, J. H. Argyris [B] and Exercise 6.57) or using the Euler forward method and a number of steps (see Section 9.6).
The relations in (6.167) to (6.176) can be directly employed to establish the strain-displacement matrices of displacement-based shell elements. However, as discussed in Section 5.4.2, these elements are not efficient because of the phenomena of shear and membrane locking. In Section 5.4.2, we introduced the mixed interpolated elements for linear analysis, and an important feature of these elements is that they can be directly extended to nonlinear analysis. (In fact, the elements were formulated originally for nonlinear analysis, and the linear analysis elements are obtained simply by neglecting all nonlinear terms.)
The starting point of the formulation is the principle of virtual work written in terms of covariant strain components and contravariant stress components. In the total Lagrangian formulation we use
\int_ {0 _ {V}} ^ {t + \Delta t} \tilde {S} _ {0} ^ {i j} \delta^ {t + \Delta t} _ {0} \tilde {\epsilon} _ {i j} d ^ {0} V = ^ {t + \Delta t} \mathcal {R} \tag {6.178}
and in the updated Lagrangian formulation we use
\int_ {t _ {V}} ^ {t + \Delta t} \tilde {S} _ {i} ^ {i j} \delta^ {t + \Delta t} \tilde {\epsilon} _ {i j} d ^ {t} V = ^ {t + \Delta t} \mathcal {R} \tag {6.179}
The incremental forms are of course given in Tables 6.2 and 6.3, but here covariant strain and contravariant stress components are employed.
As discussed in Section 5.4.2, the basic step in the MITC shell element formulation is to assume strain interpolations and to tie these to the strains obtained from the displacement interpolations.
The strain interpolations are as detailed in Section 5.4.2, but of course the interpolations are now used for the Green-Lagrange strain components ^{t+\Delta t}_{0}\tilde{\epsilon}_{ij}^{\mathrm{AS}} and ^{t+\Delta t}_{t}\tilde{\epsilon}_{ij}^{\mathrm{AS}} , where the superscript AS denotes a assumed s train. These assumed strain components are tied to the strain components ^{t+\Delta t}_{0}\tilde{\epsilon}_{ij}^{\mathrm{DI}} and ^{t+\Delta t}_{t}\tilde{\epsilon}_{ij}^{\mathrm{DI}} , obtained from the displacement interpolations (6.170) and (6.171).
The covariant strain components ^{t+\Delta t}_{0}\tilde{\epsilon}_{ij}^{\mathrm{DI}} and ^{t+\Delta t}_{t}\tilde{\epsilon}_{ij}^{\mathrm{DI}} are calculated from the fundamental expressions using base vectors,
{ } _ { 0 } ^ { t + \Delta t } \tilde { \boldsymbol { \epsilon } } _ { i j } ^ { \mathrm{DI} } = \frac { 1 } { 2 } \left( { } ^ { t + \Delta t } \mathbf { g } _ { i } \cdot { } ^ { t + \Delta t } \mathbf { g } _ { j } - { } ^ { 0 } \mathbf { g } _ { i } \cdot { } ^ { 0 } \mathbf { g } _ { j } \right) \tag {6.180}
and t+\Delta t_{t}\tilde{\epsilon}_{ij}^{DI}=\frac{1}{2}(t+\Delta t_{i}\mathbf{g}_{i}\cdot t+\Delta t_{i}\mathbf{g}_{j}-t_{i}\mathbf{g}_{i}\cdot t_{i}\mathbf{g}_{j}) (6.181)
where t + \Delta t\mathbf{g}_i = \frac{\partial^{t + \Delta t}\mathbf{x}}{\partial r_i};\qquad {}^t\mathbf{g}_i = \frac{\partial^t\mathbf{x}}{\partial r_i};\qquad {}^0\mathbf{g}_i = \frac{\partial^0\mathbf{x}}{\partial r_i} (6.182)
and we use r_1 \equiv r, r_2 \equiv s, r_3 \equiv t , and of course,
{ } ^ { t + \Delta t } \mathbf { x } = { } ^ { 0 } \mathbf { x } + { } ^ { t + \Delta t } \mathbf { u } ; \quad { } ^ { t } \mathbf { x } = { } ^ { 0 } \mathbf { x } + { } ^ { t } \mathbf { u } \tag {6.183}
Using the interpolations discussed in Section 5.4.2, with the above strain components, the MITC shell elements already presented for linear analysis in Section 5.4.2 are now obtained, including large displacement and large rotation effects. These elements satisfy the criteria of reliability and effectiveness that we enumerated in Section 5.4.2.
The elements are general since no specific shell theory has been employed. In fact, the use of the general incremental virtual work equation with only the two basic assumptions that lines originally normal to the shell midsurface remain straight and that the transverse normal stress remains zero (here, actually, the lines/directions of the director vectors are used) — is equivalent to using a general nonlinear shell theory, which in linear analysis is the 'basic shell mathematical model' identified in D. Chapelle and K. J. Bathe [C, E]. The formulation however is made even more general by relaxing these constraints and allowing a fully three-dimensional behavior, see M. Bischoff and E. Ramm [A], W. B. Krätzig and D. Jun [A], D. Chapelle, A. Ferent and K. J. Bathe [A], D. N. Kim and K. J. Bathe [A], and to model very large deformations and strains T. Sussman and K. J. Bathe [D].
6.5.3 Exercises
6.51. Consider the two-node beam element shown.
(a) Plot the displacements of the material particles corresponding to u_1^2 , u_2^2 , and \theta_2 , and evaluate the Green-Lagrange strain components corresponding to these displacements at r = s = 0 .
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x₂ 1 1 6 u₂² θ₂ tθ₂ u₁² t u₂² 2 x₁
(b) Establish the derivatives _0 u_{i,j} (i.e., \partial u_i / \partial^0 x_j ), i = 1, 2 ; j = 1, 2 , corresponding to the nodal incremental displacement and rotation variables u_1^2 , u_2^2 , and \theta_2 .
At node 1 displacements and rotations are zero; at node 2 u_{1}^{2}=0 , u_{2}^{2}=2 , \theta_{2}=10^{\circ} .
6.52. Consider the two-node beam element shown. Calculate for the degrees of freedom u_1^2 , u_2^2 , and \theta_2 the stiffness matrix 'K and nodal force vector 'F using the total Lagrangian formulation.
(a) Use the displacement method and analytical integration.
(b) Use one-point Gauss integration for the r direction.
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x2 1 1 s r t u12 u22 θ2 2 u12 x1 1 5 20 1
All nodal point displacements and rotations are zero at time t, except ^{t}u_{1}^{2}=0.1
Young's modulus E
Shear modulus G
6.53. Perform the same calculations as in Exercise 6.52 but now assume that the element is an axisymmetric shell element, with the x_{2} axis the axis of revolution.
6.54. Consider the beam element in Exercise 6.52. Calculate the stiffness matrix 'K and force vector 'F for the degrees of freedom at node 2 using the mixed interpolation of linear displacements and rotations and constant transverse shear strain (see Section 5.4.1).
6.55. Consider the four-node shell element shown. Evaluate the displacements of the particles in the element for the given nodal point displacements and director vectors at time t. Draw these displacements over the original geometry of the element.
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Original geometry of element x3 2 1 x2 10 3 4 x1 20
^ \prime u _ {i} ^ {k} = 0 \quad \text { for } i = 1, 2, 3; \quad k = 1, 2, 3
{ } ^ { t } \mathbf { V } _ { n } ^ { k } = \left[ \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right] ; \qquad k = 1 , 2 , 3
^ \prime u _ {1} ^ {4} = 0. 1; \quad^ {\prime} u _ {2} ^ {4} = 0. 1; \quad^ {\prime} u _ {3} ^ {4} = 1
\mathbf {V} _ {n} ^ {4} = \frac {1}{2} \left[ \begin{array}{c} 0 \\ - 1 \\ \sqrt {3} \end{array} \right]
6.56. Show that the expressions in (6.161) and (6.162) contain all second-order terms in \theta_{k} to obtain the increments in the director vectors. Obtain the result by a simple geometric argument and by the fact that the rotation can be expressed through the rotation matrix \mathbf{Q} , see, for example, J. H. Argyris [B], where
\mathbf {Q} = \mathbf {I} + \frac {\sin \gamma_ {k}}{\gamma_ {k}} \mathbf {S} _ {k} + \frac {1}{2} \left(\frac {\sin \frac {\gamma_ {k}}{2}}{\frac {\gamma_ {k}}{2}}\right) ^ {2} \mathbf {S} _ {k} ^ {2}; \quad \gamma_ {k} = \left(\theta_ {k 1} ^ {2} + \theta_ {k 2} ^ {2} + \theta_ {k 3} ^ {2}\right) ^ {\frac {1}{2}}
and
\mathbf {S} _ {k} = \left[ \begin{array}{c c c} 0 & - \theta_ {k 3} & \theta_ {k 2} \\ \theta_ {k 3} & 0 & - \theta_ {k 1} \\ - \theta_ {k 2} & \theta_ {k 1} & 0 \end{array} \right]
6.57. Show that the expression in (6.175) includes all second-order terms in \alpha_{k} and \beta_{k} to obtain the increment in the director vector ^t\mathbf{V}_n^k . Obtain the result by a simple geometric argument and by use of the matrix \mathbf{Q} of Exercise 6.56 but with
\gamma_ {k} = \left(\alpha_ {k} ^ {2} + \beta_ {k} ^ {2}\right) ^ {\frac {1}{2}} \quad \mathbf {S} _ {k} = \left[ \begin{array}{c c c} 0 & 0 & \beta_ {k} \\ 0 & 0 & - \alpha_ {k} \\ - \beta_ {k} & \alpha_ {k} & 0 \end{array} \right]
6.58. Calculate the covariant strain terms \tilde{t}_{0}\tilde{\epsilon}_{ij}^{DI} for the element and its deformation given in Exercise 6.56.
6.59. Use a computer program to solve for the large displacement response of the cantilever shown. Analyze the structure for a tip rotation of \pi (180 degrees) and compare your displacement and stress results with the analytical solution. (Hint: The four-node isoparametric mixed interpolated beam element performs particularly well in this analysis.)
text_image
h L M b
E = 2 0 0, 0 0 0
\nu = 0. 3 0
h = 1
L = 1 0 0
b = 1
6.60. Use a computer program to solve for the response of the spherical shell structure shown. Calculate the displacements and stresses accurately. (The solution of this structure has been extensively used in the evaluation of shell elements; see, for example, E. N. Dvorkin and K. J. Bathe [A]).
text_image
All edges of shell are hinged P Radius = 2540 Thickness = 99.45 a = 784.90 E = 68.95 v = 0.30 2a 2a
6.6 USE OF CONSTITUTIVE RELATIONS
In Sections 6.3 to 6.5 we discussed the evaluation of the displacement and strain-displacement relations for various elements. We pointed out that these kinematic relations yield an accurate representation of large deformations (including large strains in the case of two- and three-dimensional continuum elements).
The kinematic descriptions in the element formulations are therefore very general. However, it must be noted that in order for a formulation of an element to be applicable to a specific response prediction, it is also necessary to use appropriate constitutive descriptions. Clearly, the finite element equilibrium equations contain the displacement and strain-displacement matrices plus the constitutive matrix of the material (see Table 6.4). Therefore, in order for a formulation to be applicable to a certain response prediction, it is imperative that both the kinematic and the constitutive descriptions be appropriate. For example, assume that the TL formulation is employed to describe the kinematic behavior of a two-dimensional element and a material law is used which is formulated only for small strain conditions. In this case the analysis can model only small strains although the TL kinematic formulation does admit large strains.
The objective in this section is to present some fundamental observations pertaining to the use of material laws in nonlinear finite element analysis. Many different material laws are employed in practice, and we shall not attempt to survey and summarize these models. Instead, our only objectives are to discuss the stress and strain tensors that are used effectively with certain classes of material models and to present some important general observations pertaining to material models, their implementations, and their use.
The three classes of models that we consider in the following sections are those with which we are widely concerned in practice, namely, elastic, elastoplastic, and creep material models. Some basic properties of these material descriptions are given in Table 6.7, which provides a very brief overview of the major classes of material behavior.
In our discussion of the use of the material models, we need to keep in mind how the complete nonlinear analysis is performed incrementally. Referring to the previous sections, and specifically to relations (6.11), (6.78), and (6.79) and Section 6.2.3, we can summarize the complete process as given in Table 6.8.
This table shows that the material relationships are used at two points of the solution process: the evaluation of the stresses and the evaluation of the tangent stress-strain matrices. The stresses are used in the calculation of the nodal point force vectors and the nonlinear strain stiffness matrices, and the tangent stress-strain matrices are used in the calculation of the linear strain stiffness matrices. As we pointed out earlier (see Section 6.1), it is imperative that the stresses be evaluated with high accuracy since otherwise the solution result is not correct, and it is important that the stiffness matrices be truly tangent matrices since otherwise, in general, more iterations to convergence are needed than necessary.
Table 6.8 shows that the basic task in the evaluation of the stresses and the tangent stress-strain matrix is the following:
Given all stress components 'σ' and strain components 'e' and any internal material variables that we call here 'κi, all corresponding to time t,
\left\{^ {\prime} \boldsymbol {\sigma}, ^ {\prime} \mathbf {e}, ^ {\prime} \kappa_ {1}, ^ {\prime} \kappa_ {2}, \dots \right\}
TABLE 6.7 Overview of some material descriptions
| Material model | Characteristics | Examples |
| Elastic, linear or nonlinear | Stress is a function of strain only; same stress path on unloading as on loading. $^{'}\sigma_{ij} = ^{'}C_{ijrs} ^{'}e_{rs}$ linear elastic: $^{'}C_{ijrs}$ is constantnonlinear elastic: $^{'}C_{ijrs}$ varies as a function of strain | Almost all materials provided the stresses are small enough: steels, cast iron, glass, rock, wood, and so on, before yielding or fracture |
| Hyperelastic | Stress is calculated from a strain energy functional W, $\delta S_{ij} = \frac{\partial W}{\partial \delta \epsilon_{ij}}$ | Rubberlike materials, e.g., Mooney-Rivlin and Ogden models |
| Hypoelastic | Stress increments are calculated from strain increments $d\sigma_{ij} = C_{ijrs} de_{rs}$ The material moduli $C_{ijrs}$ are defined as functions of stress, strain, fracture criteria, loading and unloading parameters, maximum strains reached, and so on. | Concrete models (see, for example, K. J. Bathe, J. Walczak, A. Welch, and N. Mistry [A]) |
| Elastoplastic | Linear elastic behavior until yield, use of yield condition, flow rule, and hardening rule to calculate stress and plastic strain increments; plastic strain increments are instantaneous. | Metals, soils, rocks, when subjected to high stresses |
| Creep | Time effect of increasing strains under constant load, or decreasing stress under constant deformations; creep strain increments are noninstantaneous. | Metals at high temperatures |
| Viscoplasticity | Time-dependent inelastic strains; rate effects are included. | Polymers, metals |
and also given all strain components corresponding to time t + \Delta t and end of iteration (i - 1) , denoted as t + \Delta t \mathbf{e}^{(i - 1)}
Calculate all stress components, internal material variables, and the tangent stress-strain matrix, corresponding to t+\Delta t e^{(i-1)} ,
\left\{^ {t + \Delta t} \sigma^ {(i - 1)}, \mathbf {C} ^ {(i - 1)}, ^ {t + \Delta t} \kappa_ {1} ^ {(i - 1)}, ^ {t + \Delta t} \kappa_ {2} ^ {(i - 1)}, \dots \right]
Hence we shall assume in the following discussion that the strains are known corresponding to the state for which the stresses and the stress-strain tangent relationship are required. For ease of writing, we shall frequently also not include the superscript (i - 1) but simply denote the current strain state as {}^{t+\Delta t}e . This convention shall not imply that no equilibrium iterations are performed. However, since the solution process for the stresses
TABLE 6.8 Solution process in incremental nonlinear finite element analysis
| Accepted and known solution at time t: | stresses 'σ |
| strains 'e | |
| internal material parameters 'κ1, 'κ2, . . . |
- Known: nodal point variables
t+\Delta t\mathbf{U}^{(i-1)}and hence element strainst+\Delta t\mathbf{e}^{(i-1)} - Calculate: stresses
t+\Delta t_{\sigma}(i-1)
tangent stress-strain matrix corresponding to t + \Delta t\sigma^{(i - 1)} , denoted as \mathbf{C}^{(i - 1)}
internal material parameters ^{t+\Delta t}\kappa_{1}^{(i-1)},^{t+\Delta t}\kappa_{2}^{(i-1)},\ldots
a. In elastic analysis: the strains t + \Delta t \mathbf{e}^{(i-1)} directly give the stresses t + \Delta t \sigma^{(i-1)} and the stress-strain matrix \mathbf{C}^{(i-1)}
b. In inelastic analysis: an integration process is performed for the stresses
{ } ^ { t + \Delta t } \boldsymbol { \sigma } ^ { ( i - 1 ) } = { } ^ { t } \boldsymbol { \sigma } + \int _ { t } ^ { t + \Delta t ^ { ( i - 1 ) } } d \boldsymbol { \sigma }
and the tangent stress-strain matrix \mathbf{C}^{(i-1)} corresponding to the state t + \Delta t , end of iteration (i - 1) , is evaluated consistent with this integration process.
In isoparametric finite element analysis these stress and strain computations are performed at all integration points of the mesh in order to establish the equations used in step 3.
- Calculate: nodal point variables
\Delta \mathbf{U}^{(i)}using^{t + \Delta t}\mathbf{K}^{(i - 1)}\Delta \mathbf{U}^{(i)} = ^{t + \Delta t}\mathbf{R} - ^{t + \Delta t}\mathbf{F}^{(i - 1)}, and then^{t + \Delta t}\mathbf{U}^{(i)} = ^{t + \Delta t}\mathbf{U}^{(i - 1)} + \Delta \mathbf{U}^{(i)}
Repeat Steps 1 to 3 until convergence.
and the tangent stress-strain matrix is identical whether or not equilibrium iterations are used, we need not show the iteration superscript. All that matters is that the conditions are completely known at time t and a new strain state has been calculated for which the new stresses, internal material parameters, and the new tangent stress-strain matrix shall be evaluated.
We should note that the evaluation of the stresses and the tangent stress-strain matrix is, in our numerical evaluation of the element stiffness matrix and force vector, performed at each element integration point. Hence, it is imperative that these computations be performed as efficiently as possible.
In inelastic analysis, an integration process is needed from the state at time t to the current strain state, but in elastic analysis no integration of the stresses is required (as we employ a total strain formulation and not a rate-type formulation; see Example 6.24). In elastic analysis, the stresses and the tangent stress-strain matrix can be directly evaluated for a given strain state. Hence, in the following discussion when considering elastic conditions (Sections 6.6.1 and 6.6.2), we shall also, for further ease of writing, simply consider the strain state at time t and evaluate the corresponding stresses and tangent stress-strain matrix at that time [the same procedure is used for any time, including time t + \Delta t ].
6.6.1 Elastic Material Behavior—Generalization of Hooke's Law
A simple and widely used elastic material description for large deformation analysis is obtained by generalizing the linear elastic relations summarized in Chapter 4 (see Table 4.3) to the TL formulation:
{ } _ { 0 } ^ { \prime } S _ { i j } = { } _ { 0 } ^ { \prime } C _ { i j r s } { } _ { 0 } ^ { \prime } \epsilon _ { r s } \tag {6.184}