09 - Finite Element Formulation of Solid Continua

8/3/2019 09 - Finite Element Formulation of Solid Continua

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Finite Element Formulation of Solid ContinuaBy

S Ziaei Rad

Mechanical Engineering Department, IUT



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.



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”.



Description of Motion Consider a body with initial description of

C0.

X=(X1,X2,X3) Material coordinates



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)



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.



Deformation Gradient Tensor Consider 2 particle P and Q near each other



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.



Deformation Gradient Tensor Also,

Or in indicial notation



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



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.



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



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



Green and Almansi Strain Tensor The change in line length wrt the current body

configuration is

Almansi strain tensor



Green and Almansi Strain Tensor In indicial notation



Green and Almansi Strain Tensor In expand notation



Example Consider a block (a*b*h) where h is small wrt a

and b. Suppose that the body deform to adiamond shape



Example (cont) The coordinate mapping and its inverse is

The displacements are

The only nonzero components of Green tensor

are



Example (cont) The Almansi strain tensor



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.



Polar Decomposition To evaluate R and U

To compute U, it is necessary to write C interms of its eigenvalues and eigenvectors

Then

And



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.



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



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.



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



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.



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



Stress tensor In summary, the following relations

between different stress measure exist

J is the determinant of F



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.



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!)



Notation

C0 initial undeformed configuration C1 the last known deformed configuration

C2 the current deformed configuration (to be determined)



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.



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



Notation The following symbols in 3 configurations are used



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



Notation The displacement increment of the particle

from C1 to C2



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



Conservation of Mass A change in integration



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



Green strain tensor in various

configurations After substitution



Green incremental strain tensor The incremental strain component

Strain induced from moving from C1 to C2 It is defined



Green incremental strain tensor For geometrically linear analysis, C0=C1, C2

Thus,



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.



Updated Green strain tensor We can write



Euler strain tensor This is the strain occurred in C2 and measured

in C2

note



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



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



Relationship between various stress

tensors Then,

Deformation gradient between configuration C0 and C2



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



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



Updated Kirchhoff stress tensor According to previous relations

Since thus

R l ti



Relations

Cauchy stress and updated Kirchhoff

stress tensor

Second Piola Kirchhoff stress in C1 and C2

Relation between incremental stress



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



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



Constitutive equation In Updated Lagrangian

It can be shown that

Updated Kirchhoff stressincrement tensor

Updated Green-Lagrange strainincrement tensor

Incremental constitutive tensor wrt C1



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



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



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.



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



Total Lagrangian formulation

Next, we simplify the above equation

Since it is not a function of unknown displacements



Total Lagrangian formulationThe virtual displacements are given by

Substituting for S and E




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




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




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



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.



Total Lagrangian formulationThe total stress components are evaluated by

Where are Green-Lagrange Strain Tensor components.

( )



Total Lagrangian formulation(summary)

( )




( )






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



Updated Lagrangian formulation

The virtual strain is



Updated Lagrangian formulationwhere

Now



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



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.



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







p g g y

Updated Lagrangian formulation



summary

Finite Element Model for 2D



continuaIn the following the FE formulation based on the previous formulationis presented.

The focus is on 2D and Linear materials.

Total Lagrangian Formulation



g g

Let us introduce the following notation

The first term of the weak formulation is rewritten as




g gwhere




g g




g g

The second term can be written




g g









Where




Now let assume interpolation for both total and incremental displacements




Then we have







Substituting into the weak form one get the following relation for

the total Lagrangian formulation of 2D nonlinear continua




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.




For 2D problems







The finite element equation can be expressed in explicit form as













Updated Lagrangian Formulation



Similar to the discussion for the total Lagrangian, the FE model basedon the updated Lagrangian can be written as

where









The explicit form for the FE equation is









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09 - Finite Element Formulation of Solid Continua

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