9.1.2 Elasticity

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Subsections

9.1.2 Elasticity

9.1.2.1 Linear Elasticity

Linear elasticity requires input of the linear elastic spring stiffness.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ... * \>\>\texttt{SPRING}\>\texttt{\textit{k}}$_{r}\,$ \end{tabbing} \end{figure}$

SPRING: k is the linear elastic spring stiffness K . For translation springs, K represents force per unit elongation. For torsion springs, K is the moment per unit rotation.

9.1.2.2 Nonlinear Elasticity via Stiffness Diagram

The nonlinear elasticity model for spring elements requires input of a multilinear spring diagram.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...}}$_{r}\,$ \ldots \texttt{\textit{kn}}$_{r}\,${]} \end{tabbing} \end{figure}$

,1: SPRING specifies the diagram of stiffness: k0 ...kn (n < 50 )are the spring stiffnesses K valid until the next deformation. The last stiffness is valid until infinite deformation. Values d_ are the deformations $\delta$ (strains) until which the preceding stiffness is valid. Where $\delta$ is $\Delta$ u_x for a translation spring or $\Delta$ $\phi_{{x}}^{}$ for a torsion spring,

(file.dat)

'MATERI'
  1  SPRING 9.0
  ,1 SPRING 9.E9  -.038  9.0  0.038  9.E9

Unloading-reloading.

The special unloading-reloading model for nonlinear elasticity in spring elements requires input of multiple spring diagrams.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...}}$_{r}\,$ \ldots \texttt{\textit{kn}}$_{r}\,${]} \end{tabbing} \end{figure}$

NUMSPR: ndiag is the number of spring diagrams. ( ndiag < 50 )
ORDER: diags is a series of diagram numbers diags specifying the order in which the spring diagrams will be used at loading-unloading. The next diagram will be used each time when a switch occurs from loading to unloading or vice versa. An arbitrary selection of the specified spring diagrams may be made. Each diagram may be specified more than once. Always, the last two diagrams specified will be used till infinity. By definition, the first variation of $\delta$ (no matter whether it is increasing or decreasing) is called `loading'. When the variation of $\delta$ changes sign, the spring is said to be `unloading'. At the subsequent change in the sign of the variation of $\delta$ , the spring is said to be loading again, etc. The active diagram determines the stress increments due to the current strain increments. The stresses that are build up in the history are not relaxed if the stiffness decreases or vice versa. The default order is 1, 2, ..., ndiag.
The following data must be input for each spring diagram.
dianr: is the number of the spring diagram. Don't forget the leading comma!
SPRING: specifies the diagram of stiffness: k0 ...kn are the spring stiffnesses K₀ to K_n (n < 50 ) valid until the next deformation. The last stiffness is valid until infinite deformation. Values d_ are the deformations $\delta$ until which the preceding stiffness is valid.

(file.dat)

'MATERI'
  1
     NUMSPR  2
     ORDER   1 2 1 1
  ,1 SPRING  0.5 -1.0 2.5 1.0 4.5
  ,2 SPRING  10.5 -2.0 6.5 1.0 8.5

This example input specifies a spring with a different loading and unloading branch as shown in Figure 9.1.

**Figure 9.1:** Spring un- & reloading (example)
$\begin{figure} \setlength{\unitlength}{1cm} \begin{picture}(12.0,5.5)(0.1,0)\... ... 4.671in }% }% \centerline{\raise 5.5cm\box\graph} } \end{picture} \end{figure}$

The order of loading-unloading diagrams is 1, 2, 1, 1, 1, 1, ... .

9.1.2.3 Nonlinear Elasticity via Force-Elongation Diagram

For translation springs (SP1TR and SP2TR) you may specify nonlinear elasticity via a force-elongation diagram [Fig.9.2]. Please note the following:

To prevent ambiguity, the specified diagram must be monotonic, increasing or decreasing.

**Figure 9.2:** Force-elongation diagram for spring elements
$\begin{figure} \setlength{\unitlength}{1cm} \begin{footnotesize} \begin{picture... ...enterline{\raise 3.0cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...t{fxn}}$_{r}\,$ \texttt{\textit{duxn}}$_{r}\,${]} \end{tabbing} \end{figure}$

FDUX: specifies the force-elongation diagram. Values fx1 ... fxn (n < 30 )are the normal forces F_x . Values dux1 ... duxn are the corresponding axial elongations $\Delta$ u_x .

Next: 9.1.3 Plasticity Up: 9.1 Spring/Dashpot Behavior Previous: 9.1.1 Initial Strain Contents Index

DIANA-9.3 User's Manual - Material Library
First ed.

Copyright (c) 2008 by TNO DIANA BV.