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Journal of Minerals & Materials Characterization & Engineering, Vol. 11, No.1, pp.85-105 2012
jmmce.org Printed in the USA. All rights reserved
85
Energy Ab sorption and Strength Evaluation for Compr essed Glass Fibre
Reinforced Polyester (GRP) for Automobile Components Design in Crash
Prevention Schem e
Chukwutoo Christopher Ihueze 1 and Alfred Nwabunwanne. Enetanya 2
1Department of Industrial /Production Engineering Nnamdi Azikiwe University Awka
2Department of Mechanical Engineering Nnamdi Azikiwe University Awka
Corresponding Author: [email protected]
ABSTRACT
This paper utilized the compressive tests results to establish some critical mechanical
properties and crashworthiness parameters that may be required to design GRP composites
of polyester matrix in automobile structures. Third order polynomial function was used with
numerical methods to establish the elastic properties whish could not be established due to
sensitivity of the Monsanto tensometer used to obtain the compression results. This study
showed that the finite difference method captured the general trend of experimental solution
giving optimum value of compressive stress as 23.78MPa at strain of 0.018 and elastic limit
of 12.01MPa at 0.01 strain through finite difference analysis while the solution with third
order polynomial interpolation gave optimum compressive stress as 36.57MPa at 0.018
strain and elastic limit of 12.143MPa. Also established with compression data is the
compressive or buckling moduli of 1.2GPa. Gauss-Legendre two point rule was used to
evaluate the area under the stress-strain curve which measured the amount of energy
absorbed per unit volume of sample from where the energy absorbed at ultimate strength of
0.025J/M3- 0.22 J/M3 , energy at fracture of 0.62 J/M3- 1.62 J/M3 and the absorbed specific
work 0.001J/Kg are established.
.
Key words: viscoelastic behavior, crash parameters, energy absorption and crash work,
absorbed specific work.
1. INTRODUCTION
The prediction of damage to structures caused by accidental collision – whether to
automobiles, offshore installations or simply the packaging around an electrical appliance – is
a crucial factor in their design. This important new study focuses on the way in which
86 Chukwut oo Christophe r Ihueze Vol.11, No.1
structures and materials can be designed to absorb kinetic energy in a controllable and
predictable manner. An investigation into energy absorption requires an understanding of
materials engineering, structural mechanics, the theory of plasticity and impact dynamics.
Whilst a great deal of research has been undertaken on various aspects of these subjects, this
knowledge is diffuse and widely scattered [1].
The energy absorption capability of a composite material is critical to developing improved
human safety in an automotive crash. Energy absorption is dependent on many parameters
like fibre type, matrix type, fibre architecture, specimen geometry, processing conditions,
fibre volume fraction, and testing speed. Changes in these parameters can cause subsequent
changes in the specific energy absorption (ES) of composite materials up to a factor of 2 [2].
Composites with their high strength to weight ratio have become very important in many
technological applications such as in aerospace, automobile and medical industries. Just like
any other mechanical components in service, polymer composites are subjected to varying
mechanical forces during manufacture and use. Budiansky [3], Sridharan [4], Chung and
Weitzsman [5], Kyriakides [6] and HSU et al [7] used idealized macro-buckling mechanical
models of fibre reinforced composites to establish that the compressive strengths of
fibrecomposites subjected to compressive loads are only about 50% to 60% of their ultimate
strength in tension.
The design values for the mechanical properties of any composite system are usually
obtained from laboratory tests. These tests give valuable information on the mechanical
behavior of composite materials to a significant degree. However, the ability to understand
the response of the composites to general loading conditions or to improve their mechanical
properties requires the knowledge of the behavior of the composite on the microscope scale.
In this work buckling is considered the failure mode that governs the mec hanical behavior of
composite materials in service. The initial attempts to predict the mechanical behavior of
composite were based upon the simple theory of strength of materials.
The viscoelastic behavior of plastic composite makes the establishment of elastic range in
plastic composites difficult that experimental methods give only the short time properties of
plastic composites [8]. Foye [9] was the first to attempt the analysis of composite materials
by numerical method to obtain an inelastic solution employing a generalized plane strain
condition.
2. METHODS AND MATERIALS
The methods involve the use of the Ihueze [10] data of replicated samples of GRP composites
tested for compressive failure and the application of some numerical methods t o predict some
limiting properties of GRP composites and applying some crash evaluation relations. The
sample replication schemes are as presented in Figure 1. In the Ihueze [10] all the replications
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 87
have approximately the same pattern of behavior as designed so that in this study only sample
A was used for analysis of energy absorption and crashworthiness.
Table 1: Sample A Replication Data
Sample
variables
Sample A
A1
A2
A3
A4
Mass (kg)
0.0166
0.015
0.0157
0.0167
Length (m)
0.0829
0.0827
0.0839
0.0824
Width (m)
0.0296
0.0275
0.0279
0.0295
Thickness (m)
0.0048
0.0048
0.0048
0.0048
Area (m2)
0.00014
0.00013
0.00013
0.00014
Density (kg/m
1409
1374
1397
1431
Each of th e samples repl ications d escribed in Table 1 were subjected to compression loading
individually in the testing kit of monsanto t ensometer of Figure 1 and appropriate beam lo ad
size applied each time by application of operating handle, H. Readings of displacements
versus loads were then obtained from the autographic recorder and tabulated. The measured
readings of sample A presented in Table 2b is used in this study for analysis.
Figure 1: Line Diagram of Hounsfield Monsanto Tensometer
B
(D)
(C)
(F)
G
H
S
0
1000
2000
For ce, in kg
Spr ing
Beam
Roller
Ope rating Handle
Worm
Gear
Tension
Head
Test Piece
Operating Screw
Autographic Recording Drum
88 Chukwut oo Christophe r Ihueze Vol.11, No.1
Table 2a: Compression Force-Deformation Response Data
Sample A [10]
1
2
3
4
Deforma-
tion
(mm)
Forc e
(N)
Deforma-
tion
(mm)
Forc e
(N)
Deforma-
tion
(mm)
Forc e
(N)
Deforma-
tion
(mm)
Forc e
(N)
0.00
0
0.25
100
0.25
0
0.25
100
0.50
600
0.75
1000
1.00
1100
0.50
1000
0.75
1200
1.13
2100
2.38
1600
0.88
2000
1.25
2100
1.75
3100
2,75
1450
1.25
2900
1.50
3100
2.00
2.400
3.38
1300
1.63
2300
2.00
3600
3.00
1700
4.00
1100
2.25
2000
2.25
2600
3.50
1600
4.63
1000
2.88
1800
2.63
2200
4.25
1300
5.13
805
3.50
1700
3.00
1900
4.75
1200
5.63
500
4.25
1500
3.75
1700
5.00
1200
4.25
1500
4.75
1350
Mass (kg)
0.0166
0.015
0.0157
0.0167
Length (m)
0.0829
0.0827
0.0839
0.0824
Width (m)
0.0296
0.0275
0.0279
0.0295
Thickness (m)
0.0048
0.0048
0.0048
0.0048
Area (m2)
0.00014
0.00013
0.00013
0.00014
Density (kg/m3)
1409
1374
1397
1431
Table 2b: Compression Stress-Strain Data of Sample A [10]
A1
A2
A3
A4
Strain
(mm/mm)
Stress
(MPa)
Strain
(mm/mm)
Stress
(MPa)
Strain
(mm/mm)
Stress
(MPa)
Strain
(mm/mm)
Stress
(MPa)
0.00
0.00
0.003
0.77
0.003
0.00
0.003
0.71
0.006
4.29
0.009
7.69
0.012
8.46
0.006
7.14
0.009
8.57
0.014
16.15
0.028
12.31
0.010
14.29
0.015
15.00
0.021
23.85
0.033
11.15
0.015
20.17
0.018
22.14
0.024
18.46
0.040
10.00
0.020
16.43
0.024
25.71
0.036
13.08
0.048
8.46
0.027
14.29
0.027
18.57
0.042
12.31
0.055
7.69
0.035
12.86
0.031
15.71
0.051
10.00
0.061
6.19
0.042
12.14
0.036
13.57
0.057
9.23
0.067
3.85
0.052
10.71
0.045
12.14
0.061
8.57
0.051
10.71
0.51
9.64
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 89
3. NUMERICAL INTERPOLATIONS OF INTERMEDIATE PROPERTIES
The interpolation schemes of this section established the intermediate values of the study
such as the elastic values of the data that is needed the analysis of crashworthiness of the
material as proposed in [11]. The elastic modulus, proportionality limit and the elastic limit
are established with this interpolation schemes. The elastic modulus and limit are evaluated at
strain 0.01 [12].
3.1 Finite Difference Formulations and Polynomial Regression Method
Polynomial interpolation aids interpolation and extrapolation of data which could not be
measured due to limited sensitivity of instrument.
Analysis of a cross section of experimental data of Ihueze [10] is shown as Table 2b.By
taking a section of experimental data to the ultimate stress, a polynomial equation was
established as in figure 2 and expressed in equation (1). Our target is to capture the elastic
properties of the material being studied which will not be measured with the sensitivity range
of our instrument.
Figure 2: Third Order Polynomial Model for Finite Difference
Approximations
Supposing represents a regular partition of interval
so that following the method of [14],
90 Chukwut oo Christophe r Ihueze Vol.11, No.1
where
.
The points
are called interior mesh points of the interval
By forming a differential equation of equation (1)
By adding (1), (5) and (6)
By expressing (7) in standard linear differential equation format
By expressing (8) as
and by letting
and by replacing by their central difference approximations derived as
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 91
So that equation (4) becomes
Or by rearrangement
(10) gives the finite difference equation which is an approximation to the differential
equation of equation (9). It enables the approximation of the solution at the interior mesh
points 1, 2, of the interval [a, b].
By allowing i take on the values 1, 2, n-1 in (10) we obtain n-1 equations in the n-1
unknowns ( , , … ). Remembering that we have and since these are the
prescribed boundary conditions:
and
By observing that and in equation and
as evaluated from
= 0.002
Equation (9) reduces to
By considering the interior mesh points
For i = 1 to n-1 = 9 the following system of equations is obtained
92 Chukwut oo Christophe r Ihueze Vol.11, No.1
The right hand side of the system above is evaluated with excel spread sheet package for
evaluated with the relation of interior mesh points
expressed in as
So that since a = 0 and:
, as presented in table 3.
Table 3: Computed Mesh Point Data for Equations
I
Xi
f(xi)
0.000004f(xi)
1
0.002
167921.8
0.671687
2
0.004
130226
0.520904
3
0.006
92458.67
0.369835
4
0.008
54619.75
0.218479
5
0.01
16709.08
0.066836
6
0.012
-21273.5
-0.08509
7
0.014
-59328.1
-0.23731
8
0.016
-97454.9
-0.38982
9
0.018
-135654
-0.54262
The boundary conditions are specified from graphics of Figure 2 as
With the values of Table 3 and substituting the boundary conditions the system of equation
reduces to
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 93
The matrix equation of the system of equations becomes
=
Solving equation (30) with numerical toolkit of Chapara and Canale [13] gives values of the
function at the mesh points as:
y1 = 1.38716MPa, y2 = 3.442584MPa, y3 = 6.014278MPa, y4 = 8.950264MPa, y5 =
12.09867MPa, y6 = 15.3075MPa, y7 = 18.42493MPa, y8 = 21.29905MPa, y9 = 23.77803MPa
3.2 Newton’s Divided Difference Interpolation
The Newton’s interpolation polynomial is expressed in [13] as
Data points are used to evaluate to obtain
94 Chukwut oo Christophe r Ihueze Vol.11, No.1
Where the bracketed functions are the divided differences and the nth divided difference is
expressed as
This is a general relation for the computation of the finite divided difference of Newton’s
polynomial so that the general interpolation polynomial can be expressed as
A section of experimental data up to the ultimate stress is considered as presented in table 4.
Table 4: A section of Experimental
Data up to the Ultimate Stress
I
x
i
f(x
i
)
0
0.000
0.00
1
0.006
4.29
2
0.009
8.57
3
0.015
15.00
4
0.018
22.14
5
0.024
25.71
But for nth order polynomial, (n+1) data points are needed for complete interpolation of the
points within the interval 0 so that for this study n = 5 that is nth order
polynomial is needed.
3.2.1 Computation of finite differences
By using equation (33) and the divided differences,
0th order order divided difference
For the 1st order divided difference
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 95
For the 2nd order divided differences
So that similarly,
For the 3rd order divided differences
So that similarly
For the 4th order divided differences
So that similarly
96 Chukwut oo Christophe r Ihueze Vol.11, No.1
For the 5th order divided differences
By considering third interpolation polynomial Newton’s interpolation polynomial can be
expressed as
3.2.2 Error estimation
The truncation error is estimated with the following relation as expressed in Canale and
Chapara [13] as:
So that for n =3
3.3 Polynomial Interpolations with Numerical Tool Kit
Numerical tool kit developed by Canale and Chapara [13] was used to establish a third order
interpolation polynomial model that is of coefficient of determination 0.9898 and correlation
coefficient 0.9949 and standard error 1.62 as
This model is applied to arguments of finite difference method for comparism using excel
package and result presented as in table 5.
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 97
Table 5: Results of Interpolation Schemes Compared
X
f3nt(x)
f3nd(x)
FDM
0
0.0748
0
0
0.002
0.846154
0.42857
1.38716
0.004
2.438781
1.963952
3.442584
0.006
4.852552
4.29
6.014278
0.008
8.087334
7.090566
8.950264
0.01
12.143
10.0495
12.09867
0.012
17.01941
12.85067
15.3075
0.014
22.71645
15.1779
18.42493
0.016
29.23397
16.71507
21.29905
0.018
36.57186
17.14602
23.77803
0.024
63.50636
8.640096
25.71
4. CRASH PARAMETERS, ENERGY ABSORPTION AND CRASH WORK
EVALUATION
4.1 Crush Force Efficiency CFE
This is a very important parameter to evaluate the performance of the structure during the
crushing process. Crush force efficiency CFE is the ratio between the average crushing load
and the maximum crushing load, and can be obtained from the reasoning of Tao [15]
using values of Table 2a.
From table 2a,
Material with higher CFE will always be selected in design of energy absorbing systems.
4.2 Crashworthiness Parameters
4.2.1 Average failure load (Pav)
Average failure load Pav is a ver y important factor of the crashworthiness parameters to the
crushing energy absorbed by the structure. Material with higher Pav will always be selected
in design of energy absorbing systems.
98 Chukwut oo Christophe r Ihueze Vol.11, No.1
4.2.2 Load ratio (LR)
The main purpose for using the load ratio parameter is because it is very important in the
study of the failure modes. The load ratio LR is the ratio between the initial failure load Pi
and the maximum failure load Pmax and this can be expressed as
When Pi is taken as the value established by the interpolation scheme of table 9 then,
25.71MPa value used is the ultimate strength of table 4.
When the initial failure load Pi is of the same value of the maximum failure load the load
ratio will equal to 1 and this means that the structure initially crushed in a limited catastrophic
failure mode. But if the load ratio LR is less than 1, a matrix failure mode will be observed in
the initial crushing stage of the specimen [11].
4.3 Absorbed Energy Evaluation with Gauss-Legendre Two-Point Rule and
This is achieved by first obtaining the graphics of data of Table 6 of [10] and applying
appropriate numerical method for the areas under the stress-strain curves.
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 99
Figure 3a-d: Sample A Depiction of Area under the Stress- Strain Curve
By applying Gauss-Legendre two- point rule that will be exact for third order function the
areas under the curves are estimated as follows: By employing the polynomials of Figure 3
the areas under the curves are evaluated in order t o estimate the work or energy absorbed by
composite samples at ultimate strength and at fracture.
For Figure 3a:
The area under the curve in the finite interval [a, b] is given:
We shall attempt to determine this area using numerical approach:
Since the function is a polynomial of 3rd order, Gauss – Legendre Two – Point Rule will
be exact for evaluation of the integral [13].
The rule states that;
100 Chukwutoo Christopher Ihueze Vol.11, No.1
It all means that to apply this rule, we shall always change our finite interval [a, b] to [-1, 1]
using the transformation:
Considering the area under the curve of figure10a from the starting point up to the maximum
point A1 where the finite interval [a, b] = [0, 0.024] and
Applying the transformation (56) where [a, b] = [0, 0.024];
So that
Applying Gauss – Legendre 2 – point rule in evaluating (61)
and
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 101
so that
A1=0.025
Considering the area under the curve of (figure 3a) from the maximum point to the end A2
where the finite interval [a, b] = [0.024, 0.052]
Applying the transformation where [a, b] = [0.024, 0.052];
so that
Applying Gauss – Legendre 2 – point rule to (69)
A= A1+/A2/ =0.62
Similarly for figure 3b:
and the finite interval is [a, b] = [0.003, 0.023].
102 Chukwutoo Christopher Ihueze Vol.11, No.1
Applying Gauss-Legendre transformation on (72)
A1= 0.22, Also on the interval [0.023, 0.057] where also
and
Also, applying Gauss-Legendre transformation on (73)
A2 = -1.4, A1+/A2/ =1.62
where A1 = amount of energy absorbed or work performed on the material per
unit volume of material within the ultimate strength of material, J/m3
A1+A2 = amount of energy absorbed or work performed on the material per
unit volume of material before fracture of material, J/m3
Similar evaluations for Figures 3b and 3c are found in Table 6.
Table 6: Energy Absorption Data
Figure
A1(J/m3)
A2(J/m3)
A3(J/m3)
10a
0.025
-0.037
-
0.62
10b
0.22
-1.4
-
1.62
10c
0.1959
0.6373
-
0.83
10d
0.1531
0.8716
-0.5957
1.62
4.3.1 Total work done
The area u nder the load-d is pl acemen t cu rv e repres ent s t he t ot al en er g y abs orb ed and it can b e
calculated by multiplying the area under the stress-strain curve by the volume of the sample
so that from previous calculations, the work of the samples can be presented as in Table 7.
4.3.2 Specific energy absorption (SEA)
The specific energy absorption (SEA) is the most important factor in the design of the parts
that are need ed to r educe t heir wei ght, such as cars , airplanes and mo torc ycles, etc. The SEA
is the energy absorbed per the mass of the specimen. It can be calculated by dividing the
energy absorbed by the mass of the sample as presented in Table 8.
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 103
Table 7: Work Absorption Data
Figure
Sample
A1(J/m3)
A2(J/m3)
A3(J/m3)
(J/m3
)
Volume(m3)
W(J)
10a
A
1
0.025
-0.037
-
0.62
0.0000116
0.000007
10b
A
2
0.22
-1.4
-
1.62
0.000001
0.000017
10c
A
3
0.1959
0.6373
-
0.83
0.0000109
0.000009
10d
A
4
0.1531
0.8716
-0.5957
1.62
0.00001
0.000018
Table 8: Specific Energy Absorption (SEA)
Sample
W(J)
Mass(Kg)
SEA
A
1
0.000007
0.0166
0.0004216868
A
2
0.000017
0.015
0.0011333333
A
3
0.000009
0.0157
0.0005732484
A
4
0.000018
0.0167
0.0010778443
The average specific work is therefore calculated from table 8 as
5. DISCUSSION OF RESULTS
The tensile strength of GRP is reported to 303MPa while the compressive strength is about
50-60% of the tensile strength of material [16]. The tensile strength recorded in this study
shows that the material failed before the elastic limit of the material is reached a situation
which may be attributed to buckling in engineering. This confirms that failure of GRP may be
due to structural instability that leads to material failing before reaching the elastic limit.
The interpolations of the three numerical schemes were compared as presented in Table 5 and
figure 8 with the assertion that FDM is the better interpolation scheme for composites.
104 Chukwutoo Christopher Ihueze Vol.11, No.1
Figure 4: Results of Interpolation Schemes Compared
The graphics of this study depict the difficulty in establishing the other properties of
engineering materials as modulus of elasticity, proportionality limit, yield strength and elastic
limit.
This study showed that the finite difference method captured the general trend of analytical
solution as depicted in Figures 7 and 8. Also established by this study are the energy
absorbed at ultimate strength of 0.025J/M3- 0.22 J/M3 and energy at fracture of 0.62 J/M3-
1.62 J/M3 as depicted in table 6.
The crush force efficiency was evaluated as 77% while the average failure load and the load
ratio were evaluated as 2800N and 0.47 respectively. The load ratio 0.47 means that a matrix
failure m ode occur red, i nst ead of the cat astro phic fai lure th at occurs when t he load ratio is up
to unity.
Table 6 gave the value of absorbed energy as 0.62-1.62J/M3 while Table 7 presented the
value of the total work to lie in the range 0.000007J-0.000018J. This is an indication that the
material did not absorb much energy or that less work is done before crashing. Also table 8
gave the average value of specific work absorbed as 0.001J/Kg. This will be useful in
selecting material to be used during auto component design knowing material and limit of
energy absorption.
6. CONCLUSION
This study showed that the finite difference method captured the general trend of
experimental solution giving optimum value of compressive stress as 23.78MPa at strain of
0.018 and elastic limit of 12.01MPa at 0.01 strain through finite difference anal ysis wh ile the
solution with third order polynomial interpolation gave optimum compressive stress as
Vol.11, No.1 Energy Abs orption and S trength Evaluati on 105
36.57186MPa at 0.018 strain and elastic limit of 12.143MPa. Also established by this study
are the compressive or buckling moduli of 1.2GPa, energy absorbed at ultimate strength of
0.025J/M3- 0.22 J/M3 and energy at crash of 0.62 J/M3- 1.62 J/M3 and specific work as
0.001J/Kg. Above all material with higher CFE will always be selected in design of energy
absorbing systems.
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