# Chapter 15:Composite Materials¶

## Example 15.1 Page 552¶

In :
#initiation of variable
v_f = 40.0 # volume percent of fibre in composite
E_f= 69.0 # Modulus of elasticity of fibre in GPa
v_m = 60.0  # volume percent of matrix in composite
E_m = 3.4# Modulus of elasticity of matrix in GPa
a = 250.0 # cross sectional area in mm^2
sigma = 50.0 # Tensile stress in MPa
Fm = 1.0 # let
Ff = 13.5*Fm

#calculation
#Part A
E_cl = (v_f*E_f+v_m*E_m)/100.0

#result
print"\n Modulus of elasticity of composite is %0.0f GPa" %E_cl

# Part B"
Fc = a*sigma
Fm = Fc/(Fm+Ff)
Ff = Fc - Fm

#result
print" Force supported by m is %d" %Fm, "N  Force supported by fibre is %d N" %Ff
print "Answer in book is as Fm = 860 N and Ff = 11640. It is due to approximation"

# Part C
a_m = v_m*a/100
a_f = v_f*a/100
sigma_m = Fm/a_m
sigma_f = Ff/a_f
epsilon_m = sigma_m/(E_m*1000)
epsilon_f = sigma_f/(E_f*1000)

#result
print "Strain for matrix phase is %0.2e" %epsilon_m
print " Strain for fibre phase is %.2e. Both are identical" %epsilon_f

 Modulus of elasticity of composite is 30 GPa
Force supported by m is 862 N  Force supported by fibre is 11637 N
Answer in book is as Fm = 860 N and Ff = 11640. It is due to approximation
Strain for matrix phase is 1.69e-03
Strain for fibre phase is 1.69e-03. Both are identical


## Example 15.2 Page 553¶

In :
#initiation of variable
import math
E_gf=69  # Elasticity of glass fibre in GPa
mf_gf=0.4  #Volume percentage of glass fibre
E_pr=3.4  #  Elasticity of polyester resin in GPa
mf_pr=0.6 #Vol percentage of polyester resin

#calculation
E_ct=E_pr*E_gf/((E_pr*mf_gf)+(E_gf*mf_pr))  # Calculation of modulus of elasticity in GPa

#result
print"In transverse direction, modulus of elasticity is %.1f GPa." %E_ct

In transverse direction, modulus of elasticity is 5.5 GPa.


## Design Problem 15.1 Page 563¶

In :
#initiation of variable
from math import pi
F = 1000.0 # Force in N
L = 1.0 # length in m
del_y = 0.35 # extension in mm
d_o = 70 # Outer diameter in mm
d_i = 50 # Innrer diameter in mm
V_f_max = 0.6 # Maximum allowable fiber Volume in cm  fraction
Vf_glass = 0.945  # V_f for glass
Vf_C_standard = 0.293# V_f for carbon standard modulus
Vf_c_intermediate = 0.237# V_f for carbon intermediate modulus
Vf_c_high = 0.168 #  V_f for carbon high modulus
d_epoxy = 1.14 # density of epoxy resin in g/cm^3
d_C_sm = 1.8 # density of carbon fiber (Standard modulus) in g/cm^3
d_C_im = 1.8 # density of carbon fiber (intermediate modulus) in g/cm^3
d_C_hm = 1.8 # density of carbon fiber (high modulus) in g/cm^3
C_im_cost = 70.00 # cost of carbon fiber (intermediate modulus) in USD/kg
C_sm_cost = 35.00 # cost of carbon fiber (Standard modulus) in USD/kg
C_hm_cost = 175.00 # cost of carbon fiber (high modulus) in USD/kg
d_epoxy = 1.14 # density of epoxy resin in g/cm^3
epoxy_cost = 9.00  # cost of epoxy resin in USD/kg

#calculation
I = pi/64* (1e-12*(d_o*1e-3)**4-(d_i*1e-3)**4)
E = 4*F*L**3/(3*pi*del_y*1e-3*((d_o*1e-3)**4-(d_i*1e-3)**4)) # Required modulus of elasticity
#parta

if Vf_glass < V_f_max :
print "Glass, when embedded in epoxy matrix, meet the stipulated criteria. "

if Vf_C_standard < V_f_max :
print " Carbon (standard modulus), when embedded in epoxy matrix, meet the stipulated criteria. "

if Vf_c_intermediate < V_f_max :
print" Carbon (intermediate modulus), when embedded in epoxy matrix, meet the stipulated criteria. "

if Vf_c_high < V_f_max :
print" Carbon (high modulus), when embedded in epoxy matrix, meet the stipulated criteria."

#partb
Vc = pi*L*1e-6*(d_o**2 - d_i**2)/4
F_v_C_sm = Vc*Vf_C_standard*1e6  # Fiber Volume in cm^3 for carbon (Standard modulus)
F_m_C_sm = F_v_C_sm * d_C_sm/1000 # Fiber mass for carbon (Standard modulus) in kg
F_c_C_sm = F_m_C_sm * C_sm_cost # Fiber cost for carbon (Standard modulus) in USD
m_v_C_sm = Vc*(1-Vf_C_standard)*1e6 # Matrix Volume in cm^3 for carbon (Standard modulus)
m_m_C_sm = m_v_C_sm * d_epoxy/1000 # Matrix mass for carbon (Standard modulus) in kg
m_c_C_sm = m_m_C_sm * epoxy_cost # Matrix cost for carbon (Standard modulus) in USD
Total_c_C_sm = m_c_C_sm + F_c_C_sm # Total cost for carbon (Standard modulus) in USD
F_v_C_im = Vc*Vf_c_intermediate*1e6 # Fiber Volume in cm^3 for carbon (intermediate modulus)
F_m_C_im = F_v_C_im * d_C_im/1000 # Fiber mass for carbon (intermediate modulus) in kg
F_c_C_im = F_m_C_im * C_im_cost# Fiber cost for carbon (intermediate modulus) in USD
m_v_C_im = Vc*(1-Vf_c_intermediate)*1e6 # Matrix Volume in cm^3 for carbon (intermediate modulus)
m_m_C_im = m_v_C_im * d_epoxy/1000 # Matrix mass for carbon (intermediate modulus) in kg
m_c_C_im = m_m_C_im * epoxy_cost # Matrix cost for carbon (intermediate modulus) in USD
Total_c_C_im = m_c_C_im + F_c_C_im # Total cost for carbon (intermediate modulus) in USD
F_v_C_hm = Vc*Vf_c_high*1e6  # Fiber Volume in cm^3 for carbon (high modulus)
F_m_C_hm = F_v_C_hm * d_C_hm/1000 # Fiber mass for carbon (high modulus) in kg
F_c_C_hm = F_m_C_hm * C_hm_cost # Fiber cost for carbon (high modulus) in USD
m_v_C_hm = Vc*(1-Vf_c_high)*1e6 # Matrix Volume in cm^3 for carbon (high modulus)
m_m_C_hm = m_v_C_hm * d_epoxy/1000 # Matrix mass for carbon (high modulus) in kg
m_c_C_hm = m_m_C_hm * epoxy_cost # Matrix cost for carbon (high modulus) in USD
Total_c_C_hm = m_c_C_hm + F_c_C_hm # Total cost for carbon (high modulus) in USD

#result
print" Cost of Carbon (standard modulus) composite is:%.2f " %Total_c_C_sm # whereas Value in table is 48.50 USD
print" Cost of Carbon (intermediate modulus) composite is:%.2f " %Total_c_C_im# whereas Value in table is 71.10 USD
print"  Cost of Carbon (high modulus) composite is:%.2f " %Total_c_C_hm # whereas Value in table is 115.00 USD
print" The material of choice (i.e. least expensive)  is standard modulus carbon fiber composite; the relatively low cost per unit mass of this fiber offsets its relatively low modulus of elasticity and  required high Volume fraction."

 Carbon (standard modulus), when embedded in epoxy matrix, meet the stipulated criteria.
Carbon (intermediate modulus), when embedded in epoxy matrix, meet the stipulated criteria.
Carbon (high modulus), when embedded in epoxy matrix, meet the stipulated criteria.
Cost of Carbon (standard modulus) composite is:48.47
Cost of Carbon (intermediate modulus) composite is:71.04
Cost of Carbon (high modulus) composite is:115.84
The material of choice (i.e. least expensive)  is standard modulus carbon fiber composite; the relatively low cost per unit mass of this fiber offsets its relatively low modulus of elasticity and  required high Volume fraction.