Finite Element Method Formulation

CHAPTER 3 THEORY

3.2 Finite Element Method Formulation

For the usage of the ISM a finite element approach is employed comprised of systems of nodes and elements to derive the stress state of a component from the strains applied as given in [86].

Figure 3-1 shows a 1-Dimensional nodal system where two nodes can be seen with an attaching 𝐹1

𝑈₁ L

𝑈₁ 𝑈₂

𝐹2

𝑈₂

ΔL

Figure 3-1: 1D node displacement.(adapted from [86])

To determine the displacement of the element, denoted as U^e(x), the assumption is made that the displacement is a weighted contribution of the displacements at the ends of the bar. For the two node single element system with uni-axial loading, shown in Figure 3-1, the displacement at node 1 is given by U1, where the displacement at node 2 is given by U2. From this the displacement of the element at an arbitrary point, x, the displacement is given by Eq(4).

Eq(4) 𝑼^𝒆(𝒙) = 𝑵_𝟏𝑼_𝟏^𝒆+ 𝑵_𝟐𝑼_𝟐^𝒆= [𝑵𝟏 𝑵𝟐] [𝑼_𝟏^𝒆

𝑼_𝟐^𝒆] = 𝐍^𝒆𝐮^𝒆

The N^e terms in this case are known as the shape functions (interpolation functions) with the N^e known as the shape function matrix. The shape functions are used to interpolate from the nodal displacements the displacement of the entire element. The shape functions for this element may then be derived viewing an arbitrary point, x, on the element of length L, so that the distance from the left node may be given by Eq(5):

Eq(5) 𝑿 = 𝒙 − 𝒙_𝟏

Now for both triangles shown in figure, page 192, using the congruence of triangles we can write the following:

Eq(6) For N1: ^{𝑼𝒙,𝟏}

𝑼_𝟏 =^𝑳−𝑿

𝑳 𝒔𝒐 𝒕𝒉𝒂𝒕 𝑵_𝟏^𝒆𝑼_𝟏= [𝟏 −^𝑿

𝑳]𝑼_𝟏 Eq(7) For N2: ^{𝑼𝒙,𝟐}

𝑼_𝟐 = ^𝑿

𝑳 𝒔𝒐 𝒕𝒉𝒂𝒕 𝑵_𝟐^𝒆𝑼_𝟐= [^𝑿

𝑳]𝑼_𝟐

From this the shape functions may now be represented using isoparametric natural coordinates as defined in Eq( 8)Eq( 9) for the first and second nodes respectively, with 𝜁 the isoparametric natural coordinate correlating to the x-direction displacement.

Eq( 8) 𝑵_𝟏^𝒆= 𝟏 − ^𝑿

𝑳= 𝟏 − 𝜻 Eq( 9) 𝑵_𝟐^𝒆= ^𝑿_𝑳= 𝜻

For the derivation of the strain displacement matrix of the same system as given above, the following steps may be followed as given below. We can describe the element displacement U^e(x) as a trial (guess) displacement for the element. In this case it is not the actual displacement of the nodes as they are computed using the shape functions. This trial displacement may then be used to iteratively solve the set of solution equations until convergence is achieved, representing the approximated actual displacement of the FEM system. Using this trial displacement, the next step is to determine the associated strain in the element. For the two node system with only an axial loads and displacement, the axial strain is given by Eq( 10):

Eq( 10) Є^𝒆(𝒙) = ^{𝒅𝒖𝒆(𝒙)}

𝒅𝒙 = [^{𝒅𝑵𝟏𝒆}

𝒅𝒙 𝒅𝑵_𝟐^𝒆

𝒅𝒙] [𝒖_𝟏^𝒆 𝒖_𝟐^𝒆]

With

Eq( 11) ^{𝒅𝑵𝟏𝒆}

𝒅𝒙 = −^𝟏

𝑳

and

Eq( 12) ^{𝒅𝑵𝟐𝒆}

𝒅𝒙 = ^𝟏

𝑳

From this equation takes the form given by Eq( 13).

Eq( 13) Є^𝒆(𝒙) = 𝐁𝐮^𝐞

In this case B is called the strain displacement matrix and is given by Eq( 14), where L is the length of the element:

Eq( 14) 𝐁 = [^{𝒅𝑵𝟏𝒆}

𝒅𝒙 𝒅𝑵_𝟐^𝒆

𝒅𝒙] = ^𝟏

𝑳[−𝟏 𝟏]

This is then used to determine the strains at any given point over the element. The strain displacement matrix may then also be represented using the isoparametric natural coordinate system as given in Eq( 15)

Eq( 15) 𝐁 = [^{𝒅𝑵𝟏𝒆}

𝒅𝒙 𝒅𝑵_𝟐^𝒆

𝒅𝒙] = [^𝒅𝑵_𝒅𝜻^{𝟏𝒆𝒅𝜻}_𝒅𝒙 ^𝒅𝑵_𝒅𝜻^{𝟐𝒆𝒅𝜻}_𝒅𝒙] = ^𝒅𝜻

𝒅𝒙[^𝒅𝑵_𝒅𝜻^{𝟏𝒆 𝒅𝑵}_𝒅𝜻^𝟐𝒆] = 𝐉^−𝟏[^𝒅𝑵_𝒅𝜻^{𝟏𝒆 𝒅𝑵}_𝒅𝜻^𝟐𝒆]

Where J is the Jacobian matrix, expressed as J = ^𝑑𝑥

𝑑𝜁.for the uniaxial problem displayed. This matrix is then used to map the spatial coordinates system to the isoparametric natural coordinate system.The next step is to solve for the force displacement matrix. This is done through balancing of energies, namely, the total internal energy (Eq( 16)) and the external work done on the system.

Eq( 16) 𝐔_𝒊𝒏𝒕^𝒆 = ^𝟏_𝟐[𝒖^𝒆(𝒙)^𝑬𝒆_𝑳^𝑨_𝒆^𝒆𝒖^𝒆(𝒙)] = ^𝟏_𝟐[(𝐮^𝒆)^𝐓𝐊^𝒆𝐮^𝒆] Eq( 17) 𝐊^𝒆= ∫ 𝑬𝐀𝐁_𝟎^{𝟏 𝐓}𝐁𝑳 𝒅𝜻= ^𝑬𝑨

𝑳[𝟏 −𝟏

−𝟏 𝟏]

Where K^e is the stiffness matrix (Eq( 17)), E the elements Youngs modulus, A^e the cross sectional areaof the element and L^e the length of the element. The work done on the system may then be given by Eq( 18).

Eq( 18) 𝐖^𝒆= (𝐮^𝒆)^𝐓𝐟^𝒆

Where f^e is the nodal force with corresponding displacement as given by Eq( 19).

Eq( 19) 𝐟^𝒆= 𝐊^𝒆𝐮^𝒆

The stress of the element may then also be determined through the usage of Eq( 20).

Eq( 20) 𝛔^𝒆= 𝑬^𝒆𝐁𝐮^𝒆

For the FEM of 3-dimensional solids using hexahedral elements the following method is employed.

Firstly the solid area is discretised and allocated isoparametric natural coordinates as shown in Figure 3-2.

Figure 3-2: Hexahedral isoparametric natural coordinate system.(adapted from [86]).

With the three dimensional approach, the isoparametric natural coordinates are expanded to include three dimensions, (𝜁, 𝜂, Є). Using this, the shape functions of a hexahedral element may be written concisely as given in Eq( 21).

Eq( 21) 𝐍_𝐢= ^𝟏

𝟖(𝟏 + 𝜻𝜻_𝒊)(𝟏 + 𝜼𝜼_𝒊)(𝟏 + ЄЄ_𝒊)

Where (𝜁_𝑖, 𝜂_𝑖, Є_𝑖) represent the natural coordinates of the node Ni. For a hexahedral element the natural coordinates corresponding to figure are given in Table 3-1.

Table 3-1: Isoparametric coordinate system node numbering.

Node 𝜻 𝜼 Є

1 -1 -1 -1

2 1 -1 -1

3 1 1 -1

4 -1 1 -1

5 -1 -1 1

6 1 -1 1

7 1 1 1

8 -1 1 1

Using again equation for the elemental displacement, the displacement matrix u^e now takes the form given in Eq( 22).

Eq( 22) (𝐮^𝒆)^𝑻= [𝐔𝟏 𝐔_𝟐 ⋯ 𝐔_𝟖]

Where the displacements of the nodes Ui is now given in three dimensions as given by Eq( 23).

Eq( 23) 𝐔_𝒊= { 𝒙_𝒊 𝒚_𝒊

𝒛_𝒊} 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝟖

The shape functions are now defined as follows:

Eq( 24) 𝐍^𝒆= [𝐍𝟏 𝐍_𝟐 ⋯ 𝐍_𝟖]

Where the values for Ni are given by Eq( 25).

Eq( 25) 𝐍𝐢= [

𝑵_𝒊 𝟎 𝟎

𝟎 𝑵_𝒊 𝟎

𝟎 𝟎 𝑵_𝒊

] 𝒇𝒐𝒓 𝒊 = 𝟏, 𝟐, … , 𝟖

Using the strain displacement matrix given by equation, and incorporating the differential matrix operator given by equation, the strain displacement matrix may now be expressed by Eq( 26).

Eq( 26) 𝐋 =

[

𝝏⁄𝝏𝒙 𝟎 𝟎

𝟎 𝝏

⁄𝝏𝒚 𝟎

𝟎 𝟎 𝝏

⁄𝝏𝒛

𝟎 𝝏

⁄𝝏𝒚 𝝏

⁄𝝏𝒛

𝝏⁄𝝏𝒙 𝟎 𝝏

⁄𝝏𝒛

𝝏⁄𝝏𝒙 𝝏

⁄𝝏𝒚 𝟎 ] Eq( 27) 𝐁𝒊= 𝐋𝐍𝐢

From this the matrices may be expanded and equations used to solve for the nodal forces and stress at specified locations. To incorporate the inherent strains into the FEM model, the following approach is used. In this case an assumption of an isotropic material is again used so that a material constant matrix (i.e. stress-strain matrix) may be defined as follows and represents the material properties (Eq( 28)):

Eq( 28) 𝑪 = (𝟏+𝒗)(𝟏−𝟐𝒗)^𝑬

[

𝟏 − 𝒗 𝒗 𝒗

𝒗 𝟏 − 𝒗 𝒗

𝒗 𝒗 𝟏 − 𝒗

𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎 𝟎

𝟎 𝟎 𝟎

𝟎 𝟎 𝟎 𝟎 𝟎 𝟎

𝟏 − 𝟐𝒗 𝟎 𝟎

𝟎 𝟏 − 𝟐𝒗 𝟎

𝟎 𝟎 𝟏 − 𝟐𝒗]

Using this material constant matrix, the inherent strains may now be coupled to the FEM model using Eq( 29Eq( 32), with the definition of inherent strain discussed in subsequent sections.

Eq( 29) 𝐟^𝒆= ∫ 𝐁^𝑻𝐂ԑ^∗𝒅𝑽 ∶ 𝒘𝒊𝒕𝒉 𝒅𝑽 𝒕𝒉𝒆 𝒗𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒆𝒍𝒆𝒎𝒆𝒏𝒕 Eq( 30) 𝐮^𝒆= (𝐊^𝒆)^−𝟏𝐟^𝒆

Eq( 31) ԑ^{𝒕𝒐𝒕𝒂𝒍}= 𝐁𝐮^𝐞 Eq( 32) ԑ^{𝒆𝒍𝒂𝒔𝒕𝒊𝒄}= ԑ^{𝒕𝒐𝒕𝒂𝒍}− ԑ^∗