nonlinear theory of elasticity question using Mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg48122] nonlinear theory of elasticity question using Mathematica
• From: Anatoly Vershinin <versh1984 at mail.ru>
• Date: Fri, 14 May 2004 00:12:21 -0400 (EDT)
• References: <c7q6c7\$rph\$1@smc.vnet.net> <200405130408.AAA26744@smc.vnet.net>
• Reply-to: Anatoly Vershinin <versh1984 at mail.ru>
• Sender: owner-wri-mathgroup at wolfram.com

```Thursday, May 13, 2004, 8:08:52 AM, Narasimham G.L wrote:

NGL> Did you first solve the linear case with circular hole? Is the time
NGL> taken more due to inclusion of geometrical non-linearity? Is the
NGL> material isotropic or St.Venant type? Please send details of the
NGL> Stress function, whether plane stress/strain , PDEs, compatibility
NGL> conditions,boundary conditions etc. Using NDSolve should take only a
NGL> few seconds.I have not used the tensor package of Mathematica but
NGL> believe it should be not too difficult or time consuming.

My programs consist of two principal steps: at first it solve the
linear case, then using it program calculates nonlinear approximation
of the task. You were right about geometrical nonlinearity. If I set
semi-axes of ellipse equal, then the duration of calculation will be
about 3-5 minutes, depending on number of elements of row.  All
stresses and strains are plane. The nonlinear properties of given
material are described by Murnaghan potential. I don't use NDSolve
because there are no numerical calculus in program. All
steps(including the answer) are symbolical. Boundary conditions
consist in null stress on the boundary of elliptical hole. For solving
I use Kolosov-Mushelishvili complex potentials, expanding them to
finite row. Because of large symbolical expressions tensor operations,
integration and especially Symplify and FullSimplify,
which use these expressions are very time consuming.  By the way,
where can I get tensor package of Mathematica, as in the program
I had to wrote some tensor functions myself (I didn't find in
Mathematica nabla operator, convolution product and other operations,
which can be applied to tensors).

--
Best regards,
Anatoly                            mailto:versh1984 at mail.ru

```

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