Main
Research Contributions
Mathematical
Modeling
Beam Models:
1. Linear elastic
beam model (with D. Russell, 1996)
It is known that the classical Euler–Bernoulli beam
theory is valid only for long span (equivalently, thin) beams. In 1921, S. P.
Timoshenko proposed a dynamic beam theory with two generalized displacements;
i.e., the deflection w(x) and the
transverse shear deformation v(x).
In 1996, Gao and Russell proposed an extended beam
model which allows the shear deformation vary in y-direction, v(x,y).
Static model
Dynamical model
Finite element Simulations
extended beam model
Numerical simulations and
comparison of Euler- Bernoulli beam, Timoshenko beam with Gao-Russell beam model
Clamped beam
Clearly, these numerical
simulations show that by allowing the shear deformation, the Gao-Russell beam
is much weaker than both Euler-Bernoulli beam and Timoshenko beam models. This extended beam model can be used for
studying contact problems with friction.
2. Elasto-perfectly
plastic beam model:
The dual problem of this
model is equivalent to a quadratic minimization problem subjected to linear and
nonlinear inequality constraints given below
3. Large deformation
nonlinear elastic beam models (GAO, 1996)
wtt + k wxxxx + (ν p – a wx2 ) wxx = f(x,t)
where
k,
a and ν are given constants, p is a given axial load, f(x,t)
is a given distributed input.
This equation has at most three solutions w(x,t)
at each given (x, t), represents the two possible buckling states (upper
and down positions), and one unbuckled (unstable) position. These three
solutions could lead to chaos in dynamical vibration for certain given data k,
a, ν, the axial load p and the distributed load f(x,t)
This
equation is also equivalent to one-dimensional Landau-Ginzburg equation in
phase transitions.
In literature, this beam model is known as the Gao beam
By using this beam model, the following
interesting phenomena in chaotic vibration have been discovered in 2003.
·
Tri-chaos:
for a given periodic load f(x, t) and
a linear axial load p = po + t k, where po
and k are positive constants, the beam may
experiences three chaotic vibrations before final collapse (see the following figures).
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·
Meta
chaos: there exists a very short transition period before the chaos occurred,
see the following figures.
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Meta Chaos
Chaos
vase: A closed vision
4. Some more general
beam models are proposed in 2000
·
Fourth-order
large deformation dynamical beam model with shear effects:
where a = 3 h (1 - n2
), and the shear deformation v(x,y)
must be an old function of y. Ignoring the shear effects (v = 0 ), this beam model is reduced to the
beam model given above.
·
Second-order
large deformation dynamical beam model with shear effects:
where
the shear deformation x
could be any function of (x,y), so it can be used for studying frictional
contact problems
·
Second-order
static beam model with shear effects:
The total potential
energy is a nonconvex functional
Its canonical dual is
given by
Which is concave if the
axial stress 
is positive for all x.
Multi-scale modeling for super-conductivity materials (click here for
details)
Eigenvalue Problem on Optimal Surface (Gao and S.-T
Yau, 1991, published in Gao’s book)
Let Ω be a
domain in R2 with boundary. A regular parametrical surface S in R3 is a C2 mapping
c : W
S
The
eigenvalue problem on the optimal surface is a minimization problem
Where the contrain sets are
The criticality condition leads to the following
coupled nonlinear partial differential equations
Where 
is the Laplace-Beltrami operator, I3 = det C ,
and H(c)
is the mean curvature of the surface
And 
is the second fundamental form of the surface
with N = { Ni } being the
unit vector normal to the surface.
This problem plays an
important role in optimal design of many engineering structures. To solve this
coupled nonlinear partial differential equations is fundamentally difficult. By
using numerical simulation, some very interesting singularities on both
eigenfunction and optimal surface are discovered (see below)
Singularities
on eigenvalue problem over the optimal surface.
Some movies about finite
element simulations for dynamical post-buckling analysis of a rubber
diaphragm created by my Ph.D. student Axinte Ionita:
Optimal Shape Design for extended beam model.
Back to the home of David
Yang Gao
E-mail to me at gao at vt.edu