This document discusses modeling the elastic properties of laminated composites. It first defines composites as materials made of two or more distinct constituent materials. It then discusses various properties of composites like strength and corrosion resistance. It classifies composites based on their matrix material and reinforcement type. It also discusses volume and weight fractions. Rules of mixture are presented for calculating longitudinal and transverse modulus. Stress-strain relationships and analysis methods for orthotropic laminates are described. Forces, moments, and the inverse of the stiffness matrix for laminated composites are defined. Computer simulation of composite modeling allows for virtual testing of multiple design scenarios.

1. “Modeling of elastic properties
of laminates”
By: Harsh Kumar
18ME4109
Machine Design
National Institute Of Technology Durgapur

2. Composites
 The word “composites” means “consisting of two or more distinct parts”. Thus a material
having two or more distinct constituent materials or phases may be considered a composite
material.

3. Properties of composites
Properties of composites are strongly influenced by the properties of their constituent materials,
their distribution, and the interaction among them. The composite properties may be the volume
fraction sum of the properties of the constituents.
Its various properties are:
1) LIGHT WEIGHT
2) HIGH STRENGTH
3) STRENGTH RELATED TO WEIGHT
4) CORROSION RESISTANCE
5) HIGH-IMPACT STRENGTH
6) DESIGN FLEXIBILITY
7) DIMENSIONAL STABILITY
8) NONCONDUCTIVE
9) NONMAGNETIC
10)DURABLE
11)LOW THERMAL CONDUCTIVITY

4. Classification of composites
On the basis of matrix
material
Polymer
Matrix
Thermosets
Thermoplastic
Ceramic
matrix
Metal matrix

5. Based on
reinforcement
Fibrous
Composites
Single-layer
composites
continuous-
fiber-reinforced
Unidirectional
reinforcement
Bidirectional
reinforcement
discontinuous-
fiber-reinforced
Random
orientation
Preferred
orientation
Multi-layered
composites
Laminates
Hybrids
Particulate
Composites
Random
orientation
Preferred
orientation

6. Volume and weight fraction
Volume fraction and weight fraction are denoted by capital letters V and W respectively.
Subscript m, f and c are used to represent matrix, fiber, composite respectively.
Let
vc = volume of composite and wc = weight of composite
vf = volume of fiber and wf = weight of fiber
vm = volume of matrix and wc = weight of matrix
then volume fraction and weight fraction are given by

7. Rule of mixture
The upper bound on the composite modulus E according to rule of mixture is given by the
following relation and will represent longitudinal modulus of elasticity EL
EL=Ef*Vf+Em*Vm

8. The lower bound on the composite modulus E according to rule of mixture is given by the
following relation and will represent transverse modulus of elasticity ET
1
𝐸 𝑇
=
𝑉 𝑓
𝐸 𝑓
+
𝑉 𝑚
𝐸 𝑚
Shear modulus:
Poisson’s ratio:

9. Analysis of an orthotropic laminates
 A single layer of laminated composite material is referred
as lamina. Several laminae are bonded together to form a
structure termed as laminate. Orientation of laminae are
chosen to meet laminate design requirement. Orthotropic
material, whose behavior lies between isotropic and
anisotropic material.

10. Hooke’s law for orthotropic material
for 3D,

11. for 2D,

12. Relation between engineering constant and
element of stiffness and compliance matrix:

13. Stress strain relation for any orientation lamina:
T
x
σx
σy
y
τxy
σx
σy
L

14. Stress strain relation for orthotropic lamina for any
orientation:
Step 1:
Step 2:
Step 3:
Step 4:

15. Matrix equation can be written as

16. Analysis of orthotropic laminated composite:
Bond between two laminae is assumed to be perfect. i.e. infinitesimely
thin, not shear deformable, no slip between layer, displacement remain
continuous across the bond
Stress strain relationship:
u0
w0x
z
B
C
A
B’

17. Forces and moment in lamina:
x
y
x
y
z Nx
Ny
Nxy
Nyx
Mx
My
MxyMyx
Force in lamina
Moment in lamina
Where h = thickness of lamina

18. Forces in orthotropic laminated composite:
h
1
2
n
k
h0
h1
h2
hn
hn-1
hk-1
hk
Mid plane
Geometry of laminate

19. Moment in orthotropic laminated composite:

20. in short,
where A: extensional stiffness matrix
B: bending-extension coupling stiffness matrix
D: bending stiffness matrix
Combining both force and moment equation together, it can be written as:
(1)

21.  However, in most experiments, loads are applied and the resulting deformations are
measured, i.e. the deformations are the dependent variables, not the loads. Thus the
expressions for the middle surface extensional strains and curvatures in terms for force and
moment resultants would be convenient.
 The first step in the derivation of inverse of equation (1) is to write it in the form
(2)
(3)
and solve for Equation (2) for ℇ0
ℇ0
= 𝐴−1
N - 𝐴−1
Bk (4)
whereupon Equation (3) becomes
M= B𝐴−1N + (-B 𝐴−1B+D)k (5)
Equation (4) and (5) can be written as
(6)

22. or, ℇ0 = A*N + B*k (7)
M = H*N + D*k (8)
where B* is not equal to H*. Now solve Equation (8) for k.
k = 𝐷∗−1
M − 𝐷∗−1
H∗N (9)
and substitute in Equation (7) to get
ℇ0 = B* 𝐷∗−1M + (A* - B* 𝐷∗−1H*)N (10)
Thus,
(11)
or
Now we can calculate stress in each lamina by using following relation

23. Flow chart for
algorithm

24. Conclusion
To reduce the amount of prototype testing – Computer
simulation allows multiple “what-if” scenarios to be tested
quickly and effectively.
It is cost saving, time saving and creates more reliable better
quality designs.

25. Thank you