Analytical Vs Numerical Analysis in Solid Mechanics

Dr. Arturo A. Fuentes

Created by:Krishna Teja Gudapati

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-MechanicsCivil Engineering Manufacturing

EngineeringBiomechanics Materials ScienceMicroelectronics Nanotechnology

To know more about solid mechanics visit:http://www.engr.panam.edu/~afuentes/mechmat.htm

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 calculate2.      Identify the geometry of the solid to be modeled3.      Determine the loading applied to the solid4.      Decide what physics must be included in the model5.     Choose (and calibrate) a constitutive law that describes the

behavior of the material6.      Choose a method of analysis7.      Solve the problem

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 solids3.  3D linear elasticitity, usually solved using transforms.4.  2D (plane strain) deformation of rigid plastic solids (using slip line fields)

 

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 (Newton’s laws) for individual atoms; Molecular static's solve equilibrium equations to calculate atom positions. 

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/cm^3

Poisson’s Ratio: 0.3

Mod. Of Elasticity: 17*10^10 dyn/cm^2 =17 Gpa

Force applied: 4482216.2 dyn=100 lb

Mesh size: 50%

Thermal coefficient of expansion: 0.000027 /c

Simple Example of an Analytical Solution

Simple Example of an Numerical Solution

Taking Finite Element Method/Analysis (F.E.A) by using ALGOR SoftwareThe same problem defined in Analytical Solution and with assuming the missing data

Dimensions taken Adding Loads and Boundary conditions

StressesStrains

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

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

Stress Results

Simple Cylinder First Approx. Bone Second Approx. Bone

Cylinder with Layers Imported Approx. from a 3d Scanner

Strain Results

Simple Cylinder First Approx. Bone Second Approx. Bone

Cylinder with Layers Imported Approx. from a 3d Scanner

Nodal Displacement Results

Simple Cylinder First Approx. of Bone Second Approx. of Bone

Cylinder with Layers Imported Approx. from a 3d Scanner

FEM Resources

To know more about FEM visit:

http://www.engr.panam.edu/~afuentes/fea.htm

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.