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COMPUTER MODELLING

Linear Elasticity

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Outline

Stress-strain relationsElement types: 1D, 2D, 3DPre-processingPost-processing

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Computer ModellingHookes law

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Computer ModellingOne-dimensional elasticity

Uniform stretching:

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f

A

L

L+ L

Stress: = f/ A

Strain: = L/ L

Youngs modulus Eis a material property such that = E

f

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Computer ModellingOne-dimensional elasticity

Nonuniform stretching:

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X X+dX

x x+dx

Elongation of the differential length normalized by the original length:dx/dX 1

Displacement as a function ofX: U(X) = x(X) X

Hence, we define = dU/dX

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Computer Modelling

Shear deformation: a type of deformation that preserves volume butchanges angle

Analogous to 1D, we define the strain tensor to be

In Cartesian coordinates,

Strain in multidimensions

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ij=

1

2

ui

xj

+

uj

xi

=

1

2u+ (u)

T[ ]

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Computer Modelling

By definition, the strain tensor is symmetric. The diagonal components of represent the stretching along the coordinate axes, while the off-diagonalcomponents represent the distortion between pairs of coordinate axes in

the undeformed configuration (e.g., xy

represents the distortion betweenx-

andy-axes).

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=

xx

xy

xz

xy

yy

yz

xz

yz

zz

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Computer ModellingStress in multidimensions

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(internal) traction

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Computer Modelling

It can be shown that at a particular point, the normal n and the tractionvector t are related through a second-order tensor called the stress tensor:

The stress tensor is generally symmetric, unless there is distribution of

body moment. In Cartesian coordinates,

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t = n

=

xx

xy

xz

xy

yy

yz

xz yz zz

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Computer ModellingAlternative notations

Alternative notations for displacement components: u = ux, v = uy, w = uz

Alternative notations for shear stresses: x

= xx

, xy

= xy

, etc.

Alternative notations for shear strains: x

= xx

, xy

= 2xy

, etc.

It is also customary to write the stress and strain tensors in vector forms:

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xx

yy

zz

xy

xz

yz

xx

yy

zz

xy

xz

yz

=

ux

y+

uy

xno !

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Stress-strain relations for planar problems

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Stress-Strain relations

Linear isotropic elastic material:Eyoung modulus, Poisson ratio (-1 < < 0.5)

D =E

(1+)(1 2)

1

1

11 2

2

1 22

1 2

2

1-2

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Plane Strain Plane Strain - Axisymmetry

Plane strain Plane stress:

Axisymmetry: Geometry

Loading Material properties Bound. conditions

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Computer ModellingGeneral solution strategy

Most of the time, the displacement field u is chosen as the primaryunknown. With a certain expression ofu, we can write down the strain field and the stress field . Then we can obtain equations by requiring thesystem to satisfy the equilibrium equation

and the boundary conditions.

Next we discuss how we interpolate the displacement field with a finite

element mesh

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+ b = 0

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Element types: 1D and 2D elements

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Element types: 1D Linear bar (standard form)

if EA is constant:

kij =

L

EAdNi

dx

dNj

dxdx (i, j = 1, 2)

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Element types: 1D Linear bar (isoparametric form)

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Element types: 1D quadratic bar (isoparametric form)

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Element types: 1D quadratic bar (element stiffnes)

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Element types: 2D (isoparametric formulation)

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Element types: 2D (stiffness matrix)

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Element types: 2D (equivalent nodal forces)

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Pre-processing: linear vs. quadratic

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Post-processing

Solution from analysis: Nodal displacements u

Q1: Is the u field continuous or discontinuous?

Q2: Is the fieldcontinuous?

Q3: Is the fieldcontinuous?

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Post-processing : nodal-displacements

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Post-processing: stresses and strains

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Post-processing: nodal stresses

Computing stress at node 2:

The value of reflects the quality of the solution.