Upload
dozio
View
235
Download
0
Embed Size (px)
Citation preview
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 1/106
Finite Element Formulation of Solid ContinuaBy
S Ziaei Rad
Mechanical Engineering Department, IUT
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 2/106
Introduction In linear description of motion of solid bodies
one assumes that displacements and strains aresmall and the material is elastic.The equilibrium
equations are derived using the undeformedconfiguration.
In nonlinear analysis of beams and plates the
strain was assumed to be small and thus one canignore the geometry of the body and different
measures of strain and stress.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 3/106
Introduction When the geometry changes are significant, i.e.
displacements and strains are large, thegeometry of the body must be updated todetermine the new position x of the materialpoint X.
Thus, it is necessary to distinguish betweendifferent measure of strain and stress.
Since strain energy in an object is independent
of strain or stress measure, thus we need tointroduce the concept of “energetically pairs of strain and stress”.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 4/106
Description of Motion Consider a body with initial description of
C0.
X=(X1,X2,X3) Material coordinates
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 5/106
Description of Motion After application of load, the body deforms and
have a new configuration C with x=(x1,x2,x3)
Material or Lagrange description: the motion of
the body is referred to a reference configuration,usually C0.
Thus in Lagrange description the current
coordinates (x1,x2,x3) are express in terms of reference coordinates (X1,X2,X3) , i.e.
And the typical behavior of a variable is expresswrt material coordinates (X1,X2,X3)
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 6/106
Description of Motion In spatial or Eulerian description the motion is
described wrt the current configuration, i.e. in
terms of (x1,x2,x3). For a typical variable
Each description convoys different information. In Lagrangian Description, the focus is on material
point X. The particle X has different phi at differenttime t.
In Eulerian description, the phi is constant anddifferent material points occupied position X atdifferent time t.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 7/106
Deformation Gradient Tensor Consider 2 particle P and Q near each other
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 8/106
Deformation Gradient Tensor In configuration C0
After deformation
The displacement of material particle P and Q
Deformation Gradient Tensor F, relation
between a material line dX before deformationto the line dx after deformation.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 9/106
Deformation Gradient Tensor Also,
Or in indicial notation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 10/106
F1 1 1
1 1 1 2 3 1 1 2 3
1 2 3
2 2 2
2 2 1 2 3 2 1 2 31 2 3
3 3 3
3 3 1 2 3 3 1 2 3
1 2 3
( , , , )
( , , , )
( , , , )
x x x x x X X X t dx dX dX dX
X X X
x x x
x x X X X t dx dX dX dX X X X
x x x x x X X X t dx dX dX dX
X X X
1 1 1
1 2 3
1 1
2 2 2
2 2
1 2 3
3 3
3 3 3
1 2 3
x x x
X X X dx dX
x x xdx dX
X X X dx dX x x x
X X X
i
i I
I
i iI I
iiI
I
xdx dX
X
dx F dX
xF X
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 11/106
Deformation Gradient Tensor The deformation gradient tensor F can be
expressed in terms of displacement u, x=X+u;
The determinant of F is called Jacobian of the
motion J=det(F) If F=I, then the body is not rotated and
undeformed.
Note that F has no information about the bodytranslation.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 12/106
Green and Almansi Strain Tensor Next a general measure of deformation
independent of both translation and rotation isdiscussed.
The distance between points P and Q in C0 and
C are
Right Cauchy-Green deformation tensor
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 13/106
Green and Almansi Strain Tensor The change in line length wrt the original
configuration is
E is symmetric, i.e. E’=E
If E=0 change in square length is zero
Green strain Tensor
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 14/106
Green and Almansi Strain Tensor The change in line length wrt the current body
configuration is
Almansi strain tensor
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 15/106
Green and Almansi Strain Tensor In indicial notation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 16/106
Green and Almansi Strain Tensor In expand notation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 17/106
Example Consider a block (a*b*h) where h is small wrt a
and b. Suppose that the body deform to adiamond shape
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 18/106
Example (cont) The coordinate mapping and its inverse is
The displacements are
The only nonzero components of Green tensor
are
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 19/106
Example (cont) The Almansi strain tensor
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 20/106
Polar Decomposition Note that tensor F transform material
vector dX into spatial vector dx
Another rule for F is with help of PolarDecomposition theorem
R orthogonal rotation tensor U, V symmetric stretch tensors.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 21/106
Polar Decomposition To evaluate R and U
To compute U, it is necessary to write C interms of its eigenvalues and eigenvectors
Then
And
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 22/106
Stress Tensor The equation of motion must be derived for the
deformed configuration of the structure at timet.
However, the geometry of the deformed body
is unknown, the equation of motions must bewritten in terms of known reference
configuration.
In doing so, different measure of stress must beintroduced.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 23/106
Stress Tensor First, introduce the True stress, which is the stress
in the deformed configuration.
The stress vector
The Cauchy stress tensor is defined as thecurrent force per unit deformed area
where
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 24/106
Stress and strain measure between
configurations The determination of final configuration for a solid
body undergoing large deformation is a difficult
task. A practical way to determine the final configuration
is to apply the load incrementally.
It means that the load is applied in increments sothat the body occupies several intermediateconfigurations prior to the final configuration.
The magnitude of load increment should be such
that the computational method used is capable of predicting the deformed configuration at each loadstep.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 25/106
Stress and strain measure between
configurations In Lagrange description assume that
If the initial configuration C0 is used asreference configuration with respect to which al
quantities are measured, it is called the totalLagrangian description.
if the latest known configuration is used as
known configuration, it is called the updatedLagrangian description.
0 1 1, , , , , ,i i nC C C C C
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 26/106
Stress Tensor In order to express df in terms of a stress time
the initial undeformed area dA, we need a new
stress tensor
Where N is the unit vector normal to theundeformed area dA.
P is called first Piola-Kirchhof stress tensor and itgives the current force per unit undeformedarea.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 27/106
Stress Tensor The second Piola-Kirchhof stress tensor S is defined
as follows
As we can transform dx to dX by use of inversegradient tensor, i.e.
We can transform df (current force) to dF(transformed current force) by
P is the transformed current force per unit
undeformed area
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 28/106
Stress tensor In summary, the following relations
between different stress measure exist
J is the determinant of F
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 29/106
Energetically conjugate stress and
strain The rate of internal work done in a continuous
medium in current configuration is
Sigma is the Cauchy stress tensor and d is thesymmetric part of the velocity gradient tensor.
Sigma and d are energetically conjugate sincetheir products produces the energy stored inthe body.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 30/106
Energetically conjugate stress and
strain One can show that
Second Piola-Kirchhof stress S is conjugate with
the rate of Green strain tensor. (Prove it!)
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 31/106
Notation
C0 initial undeformed configuration C1 the last known deformed configuration
C2 the current deformed configuration (to be determined)
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 32/106
Assumptions All variables such as displacement, strain, stress
and … are known up to C1.
We wish to develop a formulation to find the
displacement fields in C2 configuration.
The deformation of the body from C1 to C2 issmall due to the load increment.
The deformation from C0 to C1 is large but
continuous.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 33/106
Notation A left superscript the configuration in which
the quantity occurred.
A left subscript the configuration with respectto which the quantity measured.
, Quantity occurred in Ci but measured in Cj
Quantity occurred and measured in the sameconfiguration (left subscript is not shown)
occurred between C1-C2 but measured in Ci
i i
i Q Q
1 2C C
i iQ Q
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 34/106
Notation The following symbols in 3 configurations are used
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 35/106
Notation When a body deform under the action of an
external load
A Particle X in C0
moves to a new position in C1
and position in C2 The total displacement of particle X in C1 and
C2
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 36/106
Notation The displacement increment of the particle
from C1 to C2
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 37/106
Conservation of Mass The relation between the mass densities
in C0, C1 and C2 can be found from conservationof mass law.
The mass of a material body remain constant
during the movement from one configuration toanother
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 38/106
Conservation of Mass A change in integration
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 39/106
Green strain tensor in various
configurations The Green strain tensors in 2 configurations C1
and C2 are defined as
In term of displacements
Note that
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 40/106
Green strain tensor in various
configurations After substitution
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 41/106
Green incremental strain tensor The incremental strain component
Strain induced from moving from C1 to C2 It is defined
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 42/106
Green incremental strain tensor For geometrically linear analysis, C0=C1, C2
Thus,
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 43/106
Updated Green strain tensor The Green strain tensor is useful for total
Lagrangian formulation.
For updated formulation we introduce the
strain wrt configuration C1 and call it
updated Green strain tensor.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 44/106
Updated Green strain tensor We can write
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 45/106
Euler strain tensor This is the strain occurred in C2 and measured
in C2
note
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 46/106
Euler strain tensor The linear part of this tensor is the infinitesimal
strain tensor
Here, the only difference is that it is measuredwrt configuration C2. For linear analysis,however, C0=C1=C2.
These strain components are energeticallyconjugate to true Cauchy stress tenmsor.
R l i hi b i
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 47/106
Relationship between various stress
tensors The Cauchy stress components in configuration
C1 an C2
Recall that second Piola-Kirchhoff stress is the
current force in C2 but transferred to C0 andmeasured per unit area in C0
Normal to unit area 0A in C0
R l i hi b i
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 48/106
Relationship between various stress
tensors Then,
Deformation gradient between configuration C0 and C2
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 49/106
Updated Kirchhoff stress tensor It is useful to defined another stress tensor in
updated Lagrangian.
Consider an infinitesimal cube containing point
P in C1
The Cauchy stress components in this point are
As the body transform from C1 to C2, the
rectangular cube deforms to a non-rectangular
tube
U d d K hh ff
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 50/106
Updated Kirchhoff stress tensor Tensor the internal force acting along the
normal and two tangential directions of each side
surface of cube in C2
The tensor can be decomposed of
Suppose are Piola-Kirchhoff
stresses in C1 and C2 configurations
Cauchy stress tensor in C1 Updated Kirchhoff stress increment tensor
Component of Kirchhoff increment stress
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 51/106
Updated Kirchhoff stress tensor According to previous relations
Since thus
R l ti
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 52/106
Relations
Cauchy stress and updated Kirchhoff
stress tensor
Second Piola Kirchhoff stress in C1 and C2
Relation between incremental stress
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 53/106
Constitutive equationMaterials for which the constitutive behavior is only a function ofthe current state of the deformation are known as elastic.
If the work done by stresses during a deformation is dependentonly on the initial state and the current configuration the materialis called hyperelastic.
For hyperelastic material there is a stored strain energy functionU0 per unit undeformed volume, such that the material elasticitytensor C is
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 54/106
Constitutive equation For fe analysis of incremental nonlinear
analysis of solid continua, it is necessaryto express stress-strain relation in
incremental form
In Total Lagrangian
Kirchhoff stressincremental tensor Green-Lagrangestrain incrementtensor
Incremental constitutive tensor wrt C0
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 55/106
Constitutive equation In Updated Lagrangian
It can be shown that
Updated Kirchhoff stressincrement tensor
Updated Green-Lagrange strainincrement tensor
Incremental constitutive tensor wrt C1
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 56/106
Principle of virtual displacement FE analysis can be done based on
◦
Displacement◦ Forces
◦ Mixed displacements and forces
• The equation then can be extracted fromprinciple of
• Virtual displacements
• Virtual forces
• Mixed virtual displacements and forces
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 57/106
Principle of virtual displacement The principle of virtual displacement says
that sum of the external virtual work done ona body and the internal virtual work stored in
the body is zero
Virtual work done byApplied forces
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 58/106
Principle of virtual displacement The virtual work equation can not be
solved since configuration C2 is unknown.
This is the main difference with linear
analysis where it is assumed the body
configuration does not change. In large deformation analysis the body
configuration is changing continuously.
The aim now is to express the virtualwork integral over a configuration which
is known.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 59/106
Total Lagrangian formulation In total Lagrangian formulation, all
quantities are measured wrt C0.
We use the following identities
Body forces wrt
C0
Boundarytractions wrt C0
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 60/106
Total Lagrangian formulation
Next, we simplify the above equation
Since it is not a function of unknown displacements
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 61/106
Total Lagrangian formulationThe virtual displacements are given by
Substituting for S and E
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 62/106
Total Lagrangian formulation
Virtual internal energy stored in the body at configuration C1
Since the body is in equilibrium at configuration C1By principle of virtual work applied to C1
and thus
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 63/106
Total Lagrangian formulation
This is the main equation for FE analysis. We just need to replace stressby strain using an appropriate stress-strain relation and ultimately bydisplacements.
Change in the virtual strainenergy due to virtual
incremental displacementbetween C1 to C2
Virtual work done byforces due to initial stress
Change in thevirtual work
done by appliedbody forces andtractions inmoving from C1
to C2
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 64/106
Total Lagrangian formulation
The equation represents the statement of virtual work for the incrementaldeformation between C1 to C2.
No approximations were made into it so far.
Next we replace into it.
This is a nonlinear equation in incremental displacement ui
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 65/106
Total Lagrangian formulationNow we assume that ui is small. This is true as the load step from C1to C2 is small.
This is the weak form for the development of finite element model basedon total Total Lagrangian formulation.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 66/106
Total Lagrangian formulationThe total stress components are evaluated by
Where are Green-Lagrange Strain Tensor components.
( )
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 67/106
Total Lagrangian formulation(summary)
( )
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 68/106
Total Lagrangian formulation(summary)
( )
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 69/106
Total Lagrangian formulation(summary)
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 70/106
Updated Lagrangian formulation In updated Lagrangian formulation, all
quantities are measured wrt C1.
We use the following identities
Body forces wrt
C1
Boundarytractions wrt C1
Updated Green-Lagrange strain tensor
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 71/106
Updated Lagrangian formulation
The virtual strain is
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 72/106
Updated Lagrangian formulationwhere
Now
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 73/106
Updated Lagrangian formulationwhere
Virtual strain energy stored in thebody at configuration C1
Since at configuration C1 the body is in equilibrium
Next we use the constitutive equation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 74/106
Updated Lagrangian formulationNext assume that the displacement ui is small andthen use the following approximation
The above equation is the weak form for the FE analysisbased on the updated Lagrangian formulation.
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 75/106
Updated Lagrangian formulationThe total Cauchy stress components are evaluated using theconstitutive equation
Where are the components of the Almansi strain tensor
Updated Lagrangian formulation summary
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 76/106
Updated Lagrangian formulation summary
Updated Lagrangian formulation summary
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 77/106
p g g y
Updated Lagrangian formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 78/106
summary
Finite Element Model for 2D
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 79/106
continuaIn the following the FE formulation based on the previous formulationis presented.
The focus is on 2D and Linear materials.
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 80/106
g g
Let us introduce the following notation
The first term of the weak formulation is rewritten as
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 81/106
g gwhere
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 82/106
g g
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 83/106
g g
The second term can be written
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 84/106
g g
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 85/106
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 86/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 87/106
Where
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 88/106
Now let assume interpolation for both total and incremental displacements
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 89/106
Then we have
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 90/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 91/106
Substituting into the weak form one get the following relation for
the total Lagrangian formulation of 2D nonlinear continua
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 92/106
The total Lagrangian formulation (and also updated) are incremental
this means that
And the stiffness matrix is the tangent stiffness matrix
The direct stiffness matrix is implicit in vector
For linear analysis
The above formulation is easily extendable to 3D problems.
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 93/106
For 2D problems
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 94/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 95/106
The finite element equation can be expressed in explicit form as
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 96/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 97/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 98/106
Total Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 99/106
Updated Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 100/106
Similar to the discussion for the total Lagrangian, the FE model basedon the updated Lagrangian can be written as
where
Updated Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 101/106
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 102/106
Updated Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 103/106
The explicit form for the FE equation is
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 104/106
Updated Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 105/106
Updated Lagrangian Formulation
8/3/2019 09 - Finite Element Formulation of Solid Continua
http://slidepdf.com/reader/full/09-finite-element-formulation-of-solid-continua 106/106