The Elastic Properties of Multi-Ply Laminates

Before starting in on the mathematical analysis it is worthwhile defining a few terms. If you are not interested in elasticity theory then skip ahead.
First, the x-y plane is the plane of the laminate, the z-direction is perpendicular to the plane of the laminate, see below:-


  • Displacements The displacements in the x, y and z directions are u,v and w respectively. It is assumed that plate displacements in the z-direction only arise from bending, there is no variation in thickness in the z-direction (ie. no through thickness strain).

  • CentreLine The centre-line is a line through the thickness of the laminate that divides the laminate vertically into two regions. When the plate is not elastically symmetric about the geometric centre line, the plane of zero bending strain will not coincide with the plane that defines the geometric centre of the plate - in fact the plane of zero bending strain moves towards the stiffer side of the plate.

  • Bending When a uniform plate bends, as shown below, there is no extension at the centerline, but on the inside of the bend (above the centre line, z is positive) there is an increasing amount of compression (negative displacement = ) as we move away from the centre line; on the outside of the bend (below the centre-line, z is negative) there is an increasing amount of tension (positive displacement = ) as we move away from the centre-line to the outer surface of the laminate. For small angles .
  • Laminate A structure made up by stacking several layers of material on top of each other and bonding the layers together. Adjacent layers generally have different elastic properties either because they are made of different materials or have different orientations.

  • Ply An individual layer of the laminate whose elastic properties are constant across the thickness of that layer. The laminate shown above has 2 plies.

Definition of Strain in a Laminate




The in-plane displacements (u and v), which are functions of position (x,y,z) within the laminate and can be related to the centre-line displacements, uo and vo and the slopes by




Now that we have the displacements, we can get the normal strain. Recall that the normal strain is defined as the fractional change in length.






Next, we substitute for u, the function , and evaluate the derivative:-




The strain term is obtained in the same way. The engineering shear strain is just the change in the angle between two initially perpendicular sides. For small strains, .









Again we can substitute for u and v then differentiate with respect to x and y. The resulting strain matrix may be written as:-




Fortunately, the above equation can be written more simply as




where is the centre-line strains and the curvatures:-


and

Definition of Force and Stress in a Laminate

When a force is applied to the edge of a laminate, all the plies of the laminate will stretch the same amount, ie. they will experience the same strain. However, the elastic properties of each ply in the laminate depend on:-
  • The Fibre and Matrix Materials
  • The Volume Fraction of Fibres
  • The Orientation of the Fibres

In other words, the stiffness of each of the plies in the direction in which the force is applied are likely to be different and since the stress in a given ply is the product of stiffness and strain, the stress in each ply will also likely be different. Since force is the product of the stress and the cross-sectional area of the ply (thickness x width) then the force acting on each ply can be determined. The sum of the forces in the individual plies must add to the applied force for equilibrium. Simplisticly:-




where Fk is the force in the kth ply of the laminate, , is the stress, tk, is the thickness of the kth layer and w, the width of the laminate. By convention, when dealing with laminates, the force is described as N, the force per unit width of the laminate or the force resultant. Mathematically, the force resultant is defined as




for the force resultant in the x-direction. The term h is the total thickness of the laminate. We can write down both the force resultants and moment resultants (force per unit width of laminate x distance) in compact form




The integration of the total laminate thickness is actually very simple since an integral is actually a sum; so we can sum the stresses in each of the individual plies. Remember, that if there is no bending then the stresses in each ply are constant. If there is bending, then the stresses in each ply will vary across the thickness of each ply.




where hk is the position of the bottom of the kth ply with respect to the centre-line of the laminate and hk+1, the position of the top of the kth ply with respect to the centre line of the laminate as shown below.




Relating force resultants {N} to strains {e} using the stiffness matrix Q

You should always remember that stress is the product of stiffness and strain no matter how complex the problem. From the previous class you should recall that



where the stiffness matrix is a function of orientation, fibre fraction and fibre and matrix materials. In a given ply, is constant hence




similarly for the moment resultants




The integrations are actually very simple since the stiffness matrices Qk, the centre-line strains and curvatures are constant in each ply, so the only variable is z, the vertical position within each ply. Therefore




and




The tensile stiffness of the laminate QL is simply [A]/h where h is the total thickness of the laminate. When B=[0], as occurs in symmetric laminates, then .

In the next section we'll demonstrate how to use MathCAD to generate the A, B and D matrices for any number of plies of aligned fibre composite, each layer having a predefined orientation and thickness. From the matrices we will show that B is in fact zero when the laminate is symmetric about the centreline and we will demonstrate how to obtain the tensile stiffness of the laminate as a function of the orientation of the laminate with respect to a uniaxial stress - in other words - just how anisotropic is the laminate.
You can skip forward and see how the bending stiffness - or flexural rigidity (EI) can be determined from the A, B and D matrices and how stresses and strains can be calculated in the laminate as whole or, more usefully, in each ply of the laminate.