Date: Thu, 28 Mar 2024 20:27:37 -0500 (CDT)
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Subject: Exported From Confluence
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In the Pre-Analysis step, we will review the following:
- Assumptions: Assumptions for classical Hertz con=
tact mechanics are discussed.
- Mathematical model: Governing equations and boun=
dary conditions, as well as additional relations will be discussed.
- FEM approach: We will discuss solution strategy =
used in solving a nonlinear problem in FEM.
This problem is a classic example of Hertz Contact Mechanics*=
em>, and hence, makes the following assumptions:
- Surfaces are continuous and non-conforming, which means that initial co=
ntact is a point or a line. In our example of sphere-plate, the initi=
al contact interface is in a form of a point.
- Strains are small.
- Solids are elastic. This means that the material response of stre=
ss and strain behaves linearly.
- Surfaces are frictionless and cannot penetrate into each other.
For analytical solution, the following additional assumption is made.
- Both objects (in our case, sphere and plate) are semi-infinitely large =
bodies (R1,R2 >> a, where a=
em> is contact radius).
Reference* S. Timoshenko and J.N. Goodier: =E2=80=9CTheory of E=
lasticity=E2=80=9D -- Chap. 13: Sect. 125, =E2=80=9CPressure betw=
een Two Spherical Bodies in Contact"
As in any static analysis, the fundamental governing equations that we m=
ust keep in mind are the stress equilibrium equations (i.e. governing e=
quation).
=
p>
In the above set of equations, it can be shown that =CF=83=
ij =3D =CF=83ji due to moment=
balance! Furthermore, we begin by making valid assumptions with rega=
rds to our problem of interest. First, we assume that there is no bod=
y force (b =3D 0) anywhere in our model.&=
nbsp; In addition, we model our problem as a plane stress problem, which me=
ans that all of the out-of-plane stress components involving =
=CE=B8-direction, can be assumed to equal zero (=CF=83=CE=
=B8 =3D =CF=84=CE=B8r =3D&nbs=
p;=CF=84=CE=B8z =3D 0). These assumptions le=
ad to the following simplifications:
=
p>
Next we list the relevant boundary conditions of our problem. The =
two types of boundary conditions, essential and natural, will be specified =
for all boundaries in our model. It must be noted that e=
ssential boundary conditions refer to displacement conditions and =
natural boundary conditions represent traction conditions. It is=
also important to observe that only one of these boundary conditions may b=
e specified at a given boundary. In addition, only one of these bound=
ary conditions is sufficient for a given boundary point.
=
p>
Along the frictionless contact interface, we specify the following bound=
ary conditions.
Here, r represents the radial position away from the axis=
of symmetry and a denotes the contact radius. We note that, due to t=
he nonlinear nature of our problem, the contact radius a will chan=
ge throughout the loading process. Even though the contact interface =
between the sphere and the surface is initially a single point, the contact=
interface will grow to become a surface as the sphere deforms.
Since the top of sphere is subjected to a point load, traction condition=
is specified at this location. We observe that since the load is bei=
ng applied to a point, traction will be infinitely large.
Along the free surface of the sphere, the boundary condition ma=
y be specified as follows.
With symmetry condition, the following boundary condition is prescribed =
along the axis of symmetry.
By identifying the governing equations and defining the boundary condition=
s, we have set up the mathematical model. We will now establish sever=
al additional relationships, which are used in the postprocessing step for =
computing stress and strain fields using these nodal displacements. T=
hese relationships are commonly referred to as the constitutive equations. =
One of these equations is the strain-displacement r=
elationship.
Second relationship is called Hooke=E2=80=99s law. For our model, we=
assume isotropic material under plane stress, and so further simplifying t=
he Hooke=E2=80=99s law results in the following equations.
=
p>
In this section, we discuss the general methods that ANSYS uses in solvi=
ng for the desired results. As the name suggests, finite element meth=
od first requires meshing the system that is to be analyzed into a finite n=
umber of elements. In ANSYS, one can manually create the mesh configu=
ration, or can alternatively let the software use a special algorithm to ge=
nerate the mesh profile, which will not be discussed in this tutorial. =
; Depending on the level of accuracy of the results that is desirable, one =
can choose to refine the mesh, so that there will be more elements near any=
region in the model. Having greater number of elements in the system=
can allow the results to converge within appropriate bounds. It shou=
ld also be noted that an element is generally comprised of multiple nodes.&=
nbsp; Configuration of the nodes in each element can vary for different ele=
ment types. As an example, an element, PLANE183, has the con=
figuration, as shown below.
Each of the eight nodes shown above can be described by displacement vec=
tors (translational and rotational components, depending on the element typ=
e) and by force vectors. Finite element method first solves for the n=
odal displacement field with the specified boundary conditions. The u=
nderlying system of equations that ANSYS solves for is shown below.
Here, [K] is also referred to as the global stiffn=
ess matrix, and contains n by n components, where n=
em> is equal to the total degrees of freedom of the system. On the ot=
her hand, {u} and {f} =
;are column matrices with n components, which represent nodal disp=
lacement fields and nodal force fields, respectively. After specifyin=
g the appropriate boundary condition in ANSYS, it then solves for these dis=
placement and force fields simultaneously.
However, in the case of our Hertz contact example, we note that the syst=
em is a highly nonlinear problem, due to the mechanical interactions betwee=
n multiple components of the system. The fact that the boundary condi=
tion at the contact interface between the sphere and the rigid plate change=
s throughout the loading process indicates that an iterative approach is ne=
cessary to converge the solutions. More specifically, we observe that=
the state of traction and the stiffness of the system depend on the displa=
cement near the contact interface.
By default, ANSYS requires that force reaction balance be satisfied with=
in a given tolerance level. If the method of linear analysis is selec=
ted, solution would most likely fail to converge for a system that contains=
a variable contact interface since only a single iteration would be perfor=
med. To overcome this issue, we introduce the Newton-Raphson (NR<=
/em>) method in solving for the solution. Given an initial guess, NR method generates a sequence of guesses that converges to a root of=
the equation. This method is based on making successive approximatio=
ns to solution using the previous value of u to determine =
K(u).
In addition to the Newton-Raphson method, other techniques can be applie=
d, in order to help convergence issues that might arise. This method,=
known as Incremental Loading technique, makes subdivisions of the load int=
o smaller steps. While increasing the number of substeps may require =
more computation, it helps to linearize the solution by making smaller load=
s, such that the residuals between iterative solution and true solution als=
o become smaller. It must be noted that these two techniques can be a=
pplied to our finite element analysis individually or can be used simultane=
ously. Using both of these, however, is most recommended, since Incre=
mental loading technique can help decrease the number of iterations require=
d to obtain a converged solution.
Once the solution has converged, the nodal displacement fields obtained =
from the final equilibrium iteration can be further used to generate the st=
rain and stress distribution at each node. In FEM, analyses, similar =
to the ones found under Mathematical Model section, are =
adopted to compute these nodal fields. However, we have to modify our=
approach slightly to take into account the fact that we now have a finite =
number of elements. This calls for a linear, first-order approximatio=
n method among the neighboring elements in computing strain distribution.&n=
bsp; In other words, the nodal strain and stress fields are calculated in t=
he following manner.
The following video shows how to create a project and set up Engineering=
Data for our problem.
Go to Step 2: Geometry
Go to all ANSYS L=
earning Modules
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