•Material nonlinearity: •nonlinear elasticity •elasto-plasticity •visco-elasticity •visco-plasticity

•Geometric nonlinearity: •elastic instability •large strain (Green Lagrange, Almansi’s )•coupled material-geometry nonlinearity (large strain)

•Contact analysis •Frictionless contact

•Dynamic structural analysis •Implicit and explicit methods of solution

Different Nonlinear Analysis

0 01

Coordinate Systems

The configuration of the body is referred to a cartesian base system,which is fixed in space. The coordinates of a particle P in the referenece

configuration are indicated by ( x , 02 3

t t t1 2 3

t 0i i

x , x ). Under action of loads after time t the particle P moves to a new position whose base position

are indicated by ( x , x , x ) then the transformation is

x x ti

0 0 0 t t t1 2 3 1 2 3

t t 0i i i

0 0 01 2 3

t t1 2

u

maps the location of P( x , x , x ) to P( x , x , x )where

u x x

Remarks

Coordinates ( x , x , x ) are called material coordinates

and coordinates ( x , x ,

t3x )are called spatial coordinates

where

Right Cauchy-Green Deformation TensorLeft Cauchy-Green Deformation Tensor

Right Stretch Tensor

0 0 0t t tX R U

X