Analytical Vs Numerical Analysis in Solid Mechanics

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Analytical Vs Numerical Analysis in Solid
Mechanics

• Dr. Arturo A. Fuentes

Created by Krishna Teja Gudapati
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Solid Mechanics

• Solid Mechanics is a collection of mathematical
  techniques and physical laws that can be used to
  predict the behavior of a solid material when
  subjected to loading.
• Engineers and scientists use solid mechanics for
  a wide range of applications, including
• Mechanical Engineering Geo-Mechanics
• Civil Engineering Manufacturing Engineering
• Biomechanics Materials Science
• Microelectronics Nanotechnology
• To know more about solid mechanics visit
• http//www.engr.panam.edu/afuentes/mechmat.htm

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Defining a Problem in Solid Mechanics

• Regardless of the field, the general steps in
  setting up a problem in solid mechanics are
  always the same
• 1.      Decide what you want to calculate
• 2.      Identify the geometry of the solid to be
  modeled
• 3.      Determine the loading applied to the
  solid
• 4.      Decide what physics must be included in
  the model
• 5.     Choose (and calibrate) a constitutive law
  that describes the behavior of the material
• 6.      Choose a method of analysis
• 7.      Solve the problem

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Choosing a Method of Analysis

• Once you have set up the problem, you will need
  to solve the equations of motion (or equilibrium)
  for a continuum, together with the equations
  governing material behavior, to determine the
  stress and strain distributions in the solid. 
  Several methods are available for this purpose.
• Analytical solution (or) Exact solution There is
  a good chance that you can find an exact solution
  for
• 1.  2D (plane stress or plane strain) linear
  elastic solids, particularly under static
  loading.
• 2.  2D viscoelastic solids
• 3.  3D linear elasticitity, usually solved using
  transforms.
• 4.  2D (plane strain) deformation of rigid
  plastic solids (using slip line fields)
•  

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Choosing a Numerical Analysis Method

• Numerical Solutions are used for most
  engineering design calculation in practice. 
  Numerical techniques include
• 1.   The finite element method This is the most
  widely used technique, and can be used to solve
  almost any problem in solid mechanics. 
• 2.  The boundary integral equation method (or
  boundary element method) is a more efficient
  computer technique for linear elastic problems,
  but is less well suited to nonlinear materials or
  geometry.
• 3.   Free volume methods Used more in
  computational fluid dynamics than in solids, but
  good for problems involving very large
  deformations, where the solid flows much like a
  fluid.
• 4.   Atomistic methods used in nanotechnology
  applications to model material behavior at the
  atomic scale.  Molecular Dynamic techniques
  integrate the equations of motion (Newtons laws)
  for individual atoms Molecular static's solve
  equilibrium equations to calculate atom
  positions. 

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Complex Bio-Mechanical Example

• Let us consider a human leg bone (Femur) with
  the following mechanical properties that are
  taken from an average healthy human being

• Mass Density 0.237 g/cm3
• Poissons Ratio 0.3
• Mod. Of Elasticity 171010 dyn/cm2 17 Gpa
• Force applied 4482216.2 dyn100 lb
• Mesh size 50
• Thermal coefficient of expansion 0.000027 /c

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Simple Example of an Analytical Solution
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Simple Example of an Numerical Solution
Taking Finite Element Method/Analysis (F.E.A) by
using ALGOR Software The same problem defined in
Analytical Solution and with assuming the missing
data
Dimensions taken
Adding Loads and Boundary conditions
Stresses
Strains
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Femur Models Taken for the Finite Element Analysis

• Simple cylinder

1st Approx. of Femur Bone
2nd Approx. of Femur Bone
Cylinder with layers
Imported Approx. from a 3d scanner
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F.E.A Structure with Loads and Boundary
Conditions Applied

• Simple Cylinder

First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from a 3d Scanner
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Stress Results

• Simple Cylinder

First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from a 3d Scanner
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Strain Results

• Simple Cylinder

First Approx. Bone
Second Approx. Bone
Cylinder with Layers
Imported Approx. from a 3d Scanner
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Nodal Displacement Results

• Simple Cylinder

First Approx. of Bone
Second Approx. of Bone
Cylinder with Layers
Imported Approx. from a 3d Scanner
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FEM Resources

• To know more about FEM visit
• http//www.engr.panam.edu/afuentes/fea.htm

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References

• Karim Khan, 2001. Physical Activity and Bone
  Health.
• John D. Curry, 1996. Bones Structure and
  mechanics.
• Bourne, Geoffrey H., The biochemistry and
  physiology of bone.
• Kardestuncer, H., 1987. Finite Element Handbook,
  McGraw-Hill, New York.
• Nikishkov, G.V., 1998. Introduction to the Finite
  Element Method, unpublished lecture notes,
  University of Arizona, Tucson, AZ.
• Segerlind, L. J., 1984. Applied Finite Element
  Analysis, John Wiley and Sons, New York.