ClearAll["`*"]

Introduction to 
Classical Lamination Theory 
using Mathematica; Part - 1

by  
 Zafer Gurdal © Copyright 1993, 1999
 Department of Engineering Science and Mechanics 
 Virginia Polytehnic Institute and State University
 Blacksburg, VA 24061-0219

This notebook is intended to be used as a tutorial to demonstrate the use of Mathematica and Mathematica programing to calculate the properties of a laminated composite plate based on Classical Lamination Theory.

Initializations

Following lines establish the environment for the tutorial and load the Mathematica  package that has the various modules needed for the calculations.  
The first line in the next cell is to display two dimensional lists as matrices, and the next one sets the default font for the displayed text.

$PrePrint = If[MatrixQ[#1], MatrixForm[#1], #1]&; $DefaultFont = {"Times-Italic", 12};

Next we need to load the Mathematica  package clt.m  (or CLT.m) from the appropriate directory.   
	
To find the current directory Mathematica  expects to load from use

Directory[]

To find the names of the files in the current directory  use

FileNames[]

To load a program file use   <<   with appropriate path leading to the directory in which the file you want to load resides from the  current directory

<< "clt.m"

In order to find the operations that are currently supported by the clt.m package use;

?CLT

Qmatrix,		   Nu12,		    InvariantU,   	
 		Tmatrix, 		   Qbarmatrix, 	               MaterialProperties,
 		Amatrix,		   Bmatrix, 		    Dmatrix,		
 
The functions of these modules, and the input required to call them will be explained in the following sections.  However, for those who would like to know about these modules before getting into the tutorial, information can be obtained by typing a question mark followed by the name of the module.

?MaterialProperties

Properties of Unidirectional Laminae

Composite laminates are build by stacking layers of unidirectional fiber reinforced material with various orientations together and consolidating them into a plate-like material.  The consolidation process usually requires high temperature and pressure, and often referred to as the curing process.  Since the individual layers are the building blocks of a laminate, we first start with the properties of a layer.  A unidirectional fiber reinforced layer is also reffered to as a ply  or a lamina  (plural plies or laminae) in the composite materials literature.  
	
	For a unidirectional fiber reinforced layer, there are two perpendicular planes of symmetry that define two principal axes of material properties.  These principal axes correspond to the direction of the fibers and a direction transverse to the fibers and are denoted by subscripts 1 and 2, respectively.  Such layers are referred to be orthotropic, and the 1-2 axes are the principal axes of orthotropy (sometimes referred to as the principal material directions). 
	
	Elastic mechanical properties of a fiber reinforced layer are characterized by four independent material properties that relate the vector of stresses σ = {(Subscript[σ, 1]),  (Subscript[σ, 2]),  (Subscript[τ, 12])}^T to the corresponding strain vector ϵ = {(Subscript[ϵ, 1]),  (Subscript[ϵ, 2]),  (Subscript[τ, 12])}^T  through the matrix relation  σ =  Qmatrix ϵ, where

?Qmatrix

Qmatrix[{one, two, onetwo, nuonetwo}]

?Nu21

Nu21[{e1, e2, g12, nu12}]

Qmatrix[{e1, e2, g12, nu12}]

For a given set of elastic material properties, a set of five numbers  InvariantU = {U1, U2, ....., U5} is sometimes used to characterize the properties of an orthotropic layer in the principal material directions.

?InvariantU

uvect = InvariantU[{e1, e2, g12, nu12}]

uvect[[1]]

{u1, u2, u3, u4, u5} = uvect

u1

Determine the reduced stiffness matrix and the material property vector of T300/5208 graphite/epoxy lamina. 

For some of the more commonly used material systems, elastic material properties are provided in the program.  For example,

MaterialProperties["T300/5208"]

MaterialProperties["T300/5208", MetricUnits -> True]

{e1, e2, g12, nu12} = MaterialProperties["T300/5208"]

Qmatrix[{e1, e2, g12, nu12}]

Qmatrix[MaterialProperties["T300/5208"]]

Nu21[MaterialProperties["T300/5208"]]

uvect = InvariantU[MaterialProperties["T300/5208"]]

e1 =. ; e2 =. ; g12 =. ; nu12 =. ; uvect =. ; u1 =. ; u2 =. ; u3 =. ; u4 =. ; u5 =. ;