Microstructural Flaws in Bone Associated with Bone
Remodeling
FEM Modeling of Instrumented IndentationMAE 5700: Finite Element
Analysis for Mechanical and Aerospace Design
Joseph Carlonia, Julia Chenb, Jonathan Mathenyc, Ashley
TorrescaMaterials Science PhD program,
[email protected] Engineering PhD program,
[email protected] Engineering PhD program,
[email protected], [email protected]
Introduction to instrumented indentationA special form of
indentation hardness testing where load vs. displacement data is
collected continuously
The resulting load-displacement data can be used to determine
the plastic and elastic properties of the material
Commonly used to test the elastic properties of a material,
especially at a small scale nanoindentation
2Because of this advantage, instrumented indents are commonly
used... Especially at the small scale... i.e. nanoindentation
2A Real Nanoindentation Experiment1. Load Application2.
Indentation3. Load Removal
3Here is a schematic of a real nanoindentation experiment. The
instrument applies a load, the indenter penetrates the surface, and
then the load is removed. The collected load-displacement data is
usually plotted like so (load on y vs. displacement on x). After
some critical load, the material plastically deforms (that is
doesnt return to its initial displacement), but elastic properties
can still be calculated from a linear fit to the initial unloading.
Alternatively, you can try to to stay below the critical load for
plasticity.3Nanoindentation EquationsThe reduced modulus of contact
between two materials is a function of the Youngs moduli:
Sneddons equation relates the reduced modulus of a contact to
the contact stiffness and contact area:
4Sneddon, 1948 W.C. Oliver and G.M. Pharr (1992). For contact
between 2 different materials, the reduced modulus is a function of
the materials Youngs moduli.Sneddons equation (originally derived
in 1948, and applied to nanoindentation in the early 90s) relates
the reduced modulus to a measured contact stiffness and contact
area4Motivation / Problem StatementSneddons equation was derived
for contact between a rigid indenter and a semi-infinite half
spaceWe want to model the elastic portion of an indentation in
ANSYS so that we can vary dimensional parameters to see how they
affect the accuracy of Sneddons equation
2D AxisymmetricPEi, viEs, vshw5The analytical solution for
indentation was derived for...In practice, this is not possible, so
we want to model it using FEA and see how close we can approximate
the analytical solution by varying dimensional parameters and
boundary conditions.5Solid Body ContactAssume: strains are small,
materials are elastic, surfaces are frictionlessContact is a
changing-status nonlinearity. The stiffness, depends on whether the
parts are touching or separatedWe establish a relationship between
the two surfaces to prevent them from passing through each other in
the analysis termed, contact compatibility
ANSYS Academic Research, Release 14.5, Help System, Introduction
to Contact Guide, ANSYS, Inc.6Normal Lagrange FormulationAdds an
extra degree of freedom (contact pressure) to satisfy contact
compatibility Contact force is solved for explicitly instead of
using stiffness and penetration Enforces zero/nearly-zero
penetration with pressure DOFOnly applies to forces in directions
Normal to contact surfaceDirect solvers are used
ANSYS Academic Research, Release 14.5, Help System, Introduction
to Contact Guide, ANSYS, Inc.7Penalty-Based FormulationsConcept of
contact stiffness knormal is used in bothThe higher the contact
stiffness, the lower the penetration As long as xpenetration is
small or negligible, the solution results will be accurate
The Augmented Lagrange method is less sensitive to the magnitude
of the contact stiffness knormal because of (pressure)
Pros (+) and Cons (-)ANSYS Academic Research, Release 14.5, Help
System, Introduction to Contact Guide, ANSYS, Inc.88ANSYS Detection
MethodAllows you to choose the location of contact detection in
order to obtain convergence Normal Lagrange uses Nodal Detection,
resulting in fewer pointsPure Penalty and Augmented Lagrange use
Gauss point detection, resulting in more detection points
ANSYS Academic Research, Release 14.5, Help System, Introduction
to Contact Guide, ANSYS, Inc.9ANSYS Contact StiffnessNormal
stiffness can be automatically adjusted during the solution to
enhance convergence at the end of each iteration The Normal Contact
Stiffness knormal is the most important parameter affecting
accuracy and convergence behavior Large value of stiffness gives
more accuracy, but problem may be difficult to convergeIf knormal
is too large, the model may oscillate, contact surfaces would
bounce off each other
ANSYS Academic Research, Release 14.5, Help System, Introduction
to Contact Guide, ANSYS, Inc.10Nonlinear Finite Element
Approach
Newton-Raphson Iterative MethodLoading Incrementation
ProcedureBecker, A.A. An Introductory Guide to Finite Element
Analysis. p.109-125.
11Initial problem set-upMaterialsIndenter- DiamondYoungs
Modulus=1.14E12 PaPoissons Ratio=0.07
Tested Material- CalciteYoungs Modulus=7E10 PaPoissons
Ratio=0.3
Both Materials Type Isotropic Elasticity
12
Initial problem set-upAxisymmetric ModelBoundary ConditionsFixed
displacement (in x) along axis of symmetryFixed support on bottom
edge of materialLoadingPressure (1E8 Pa) applied normal to top edge
of indenter 13Automated Calculations
14
ANSYS Default Mesh (10 divisions)
Quadrilateral Elements15ANSYS Default Results (-13.4% error)
16Refined Mesh (160 divisions)
Quadrilateral Elements17Refined Results (2.74% error)
18Mesh Convergence
The magnitude of the error converges Now we change other
parameters19Normal Lagrange(9.29% error, 5e-17 m penetration)
20Augmented Lagrange(2.74% error, 1e-9 m penetration)
21Final setupContact Type: Frictionless Target Body:
indenterContact Body: materialBehavior: SymmetricContact
Formulation: Augmented LagrangeUpdate Stiffness: Each
IterationStiffness factor: 1Auto time step: min 1, max 10Weak
springs: off
22PressureToo high of a pressure increases the error23Sneddons
equation is derived for purely elastic, small deformations. We
chose a pressure of 1e8 based on experimental results. As the
pressure increases, we will eventually have large deformations that
should result in plasticity. Therefore, we expect the error to
increase after some critical pressure.23Dimension of materialToo
small of a sample increases the error24Sneddons equation is derived
assuming contact with a semi-infinite half space. As the material
size decreases, we will eventually have a geometry that cannot be
accurately approximated as infinite. Therefore, we expect to see an
increase in the error below some critical material size.24Different
modulusTesting a high modulus material increases the
error25Sneddons equation was initially derived for contact by an
perfectly rigid indenter. The concept of reduced modulus was
introduced as a correction for indenters with some finite
stiffness. As we increase the stiffness of the material being
indented, eventually the ratio between the stiffness of the
material and that of the indenter becomes too high to be accurately
corrected. Therefore, we expect the magnitude of the error to
increase with increasing material modulus.25ConclusionIndentation
can be accurately modeled using ANSYS and a well-refined mesh
The validity of Sneddons equation has been explored:Lower
pressure More accurateLarger sample More accurateMore compliant
sample More accurate26Questions?27