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A STUQY OF A SFECEAL smssscsmam
EQUATEON WETH A3323L£€ATSO§£
.TO SEVERAL METALS

Thosés {at the 3091'” 95 5%.. S.
MEGflGAN STATE CGLLEGE

gain: Ewame {gemsky
€955

THESIS

This is to certify that the
thesis entitled
A STUDY OF A SPECIAL STRESS-STRAIN EQUATION
WITH APPLICATION TO SEVERAL METAIS

presented by
JOHN JEROME LENOSKY

has been accepted towards fulfillment
of the requirements for

MASTER OLSEENQE. degree in WCWICS

(9&0N—wJ—o

Major professor

 

Date May 2?. 1955

0-169

 

 

”33$. ,, ..—. ~— 1—132..- _3:.__.

A STUDY OF A SPECIAL STRESS-STRAIN EQUATION
WITH APPLICATION TO SEVERAL METALS

By
John Jerome_§gnosky

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Applied Mechanics

1955

CV

Acmoumncmmm

The author wishes to express his sincere
appreciation to Dr. Charles 0. Harris of the
Department of Applied fiechanics, for his guidance
and invaluable assistance in the preparation of

this thesis.

CO
! 7,

L3

£42):
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A STUDY OF A SPECIAL STRESS-STRAIN EQUATION
‘WITH APPLICATION TO SEVERAL METALS

By
John Jerome Lenosky

AN ABSTRACT

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
fbr the degree of

MASTER OF SCIENCE

Department of Applied Mechanics
1955

Approved by («SLJk£Q7(D.r%£a*Q*¢O

 

 

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John Jerome Lenosky
l

The problem.undertaken, in this study, is the formula-
tion of a special stress-strain equation, which is applicable
to a wide range of materials, and the development of an easy
and accurate method of fitting this equation to the stress-
strain curve of each material.

The equation,¢7'= e tanh (A/e)N has been formulated
by the author and is proposed as a suitable representation
of the relationship existing between stress and strain. In
this equation, e is the unit strain and. 0' is the ratio of
stress to the modulus of elasticity of the given material,
or S/E. E, the modulus of elasticity, is the slope of the
actual stress-strain curve at the origin. A and N are para-
meters to be determined for each stress-strain curve.

The parameters, A and N, can be determined for each
stress-strain curve, by writing the proposed equation in
the form

log (l/e) a log (l/A) 4- ~33- log tanh'.1 (‘T/e).

This is an equation of the first degnee in log (l/e) and

log tanh.’1 GZ/e), and with these functions as coordinates
will plot as a straight line, if the values of e and 0'

are taken from.a curve of the equation, 0' I e tanh (A/e)N.
The straight line will have a slepe of l/N, and will inter-
cept the line representing zero values of log tanh."l (cr/e)
at values of log (l/A) 3 log (l/e). when values of e and

 

c71are
strai

suitic

log 1
from
and l
N is

plot;

direc'

«was

severe

the f(

John Jerome Lenosky
2

(7 are taken from a stress-strain curve to be fitted, a
straight line is drawn which by eye gives the best approxi-
:mation to the plot of log (l/e) against log tanh."1 (073).

A special coordinate paper was constructed so that
log (l/e) can be plotted against log tanh'l (Q79) directly
from.values of e and.(07e), without first calculating log (l/e)
and log tanh."1 (075). On this coordinate paper the parameter I
N is the tangent of the angle formed by the straight line
plot and the horizontal axis. The parameter A can be read
directly from the paper.

Using the special coordinate paper, the proposed
equation was fitted to typical stress-strain curves of
several metals. A good fit to each curve was obtained.

The stress-strain equation,<T 8 e tanh (A/eln, has
the following desirable properties: A

(l) The parameters, A and N, can be easily and accur-

, ately determined, for a good fit to experimental
data, by using the special coordinate paper.

(2) The curve passes through the origin.

(3) The slope of the curve at the origin can be shown

to be equal to the modulus of elasticity.

(h) The equation is applicable to a wide range of

materials.

TABLE OF CONTENTS

‘1!
p
(n
0

STATEMENT OF THE PROBLEM . . . . . . . . . . . . . .
I. HISTORICAL BACKGROUND . . . . . . . . . . . . . .
The Empirical Stress-Strain Equation . . . . . .
Properties Desired in any Stress-Strain Equation
Types of Stress-Strain Equations Proposed . . .
n. THEPROPOSEDEQUATION
The Mathematics of the Proposed Equation . . . .
Construction of the Special Coordinate Paper . 0
III. FITTING THE PROPOSED EQUATION TO A TYPICAL CURVE.

Procedure Followed in the Determination of the
Parameters . . . . . . . . . . . . . . . . .

IV. EQUATION (l) FITTED TO TYPICAL STRESS-STRAIN

CURVESOFDIFFERENTMATERIALS..........
V. CONCLUSIONS . . . . . . . . . . . . . . . . . . .
BIBLIOGRAPHY,....................

QUIUIUNNNH

3.:
H

ll

17
22

STATEMENT OF THE PROBLEM

The problem undertaken, in this study, is the formu-
lation of a special stress-strain equation, which is
applicable to a wide range of materials, and the develop-
ment of an easy and accurate method of fitting this equation

to the stress-strain curve of each material.

CHAPTER I

HISTORICAL BACKGROUND

The Empirical Stress-Strain Equation
In certain calculations such as plastic bending and

 

column analysis, it is desirable to have a mathematical
expression defining the relationship existing between the
two variable quantities, stress and strain.

When the values of stress and strain obtained from.a
compression or tension test are plotted on a coordinate
system, values of stress being plotted as the ordinates
against values of strain as the abscissae, a smooth curve
can be drawn through or near the points. The fact that
this smooth curve can be drawn leads one to suspect that a
mathematical equation exists which closely approximates the
relationship existing between stress and strain.

Properties Desired in an1;Stress-Strain Equation

A suitable stress-strain equation should possess cer-

 

tain properties in addition to being the equation of a

curve closely approximating the actual stress-strain curve.

The properties required and desired of any stress-strain

equation are listed below.1

 

1 ‘William.R. Osgood, Stress-Strain Formulas, Journal of
the Aeronautical Sciences, vol. 13, (January, 19E6), p. AS.

(a’

0))
(cl

(6.)

(a)
(f)

(g)

Simplicity. A suitable stress-strain equation
should be as simple as possible, so that its use
in calculations is not too laborious. Equations
having more than two parameters are undesirable
for this reason.

The curve should pass through the origin.

The slope of the curve at the origin should be
equal to the modulus of elasticity.

The equation should be as general as possible.
Equations having only one parameter are not adapt-
able to a wide range of materials.

The equation should give physically possible
values of stress for all values of strain.

The equation should lead to integrals of closed
form» in plastic bending and column analysis.
The parameters should be easily and accurately

determinable for a good fit to experimental data.

Types of Stress-Strain Eguations Proposed

A number of stress-strain equations have been proposed,

but none have been entirely satisfactory on all of the points

listed above. Hooke's law, e ==03 was proposed in 1678,

and is the earliest stress-strain relation.2 In this equation,

e is the unit strain and cr is the ratio of stress to the

modulus of elasticity, 8/3. This equation gives an accurate

 

2

Ibid.. p. LL60

~' lrelationship between stress and strain as long as the propor-
tional limit of the material is not exceeded.
Polynomial expressions of the following types have

been proposed.3

\]

e = cr + as“ 1 Tag 1* Scrip-p.
0-: e +b€2+ ce" + de‘+-~-°

These equations should fit practically any curve if a suf-
ficient number of terms are taken, but the parameters are
then difficult to determine. Some of the other equations
proposed are derived from these polynomials, retaining only
part of the terms.
A Ramberg and Osgoodh proposed the equation,

9 = 0’ + K 0’ "
where K and n are parameters which can be determined
easily, by plotting on logarithmic paper. This equation
satisfied the requirements of a suitable stress-strain.rela-
tion reasonably well and has been used extensively.

Other equations have also been proposed, but each
possesses some undesirable properties. The parameters in
some are difficult to determine; others are applicable to
only a limited number of materials or fail to meet all the

requirements of a suitable stress-strain equation.

 

3 ____Ib1dos P0 14-50
1* Ibid., p. m.

CHAPTER II

THE PROPOSED EQUATION

_T1_1_e Mathematics of the Proposed Equation

The equation,
N
a' = e tanh (A/e) . (1)

has been formulated by the author and is proposed as a
suitable representation of the relationship existing be-
tween stress and strain. .In this equation, e is the unit
strain, 0' is the ratio of stress to the modulus of elas-
ticity, S/E, and the parameters to be determined for each
stress-strain curve, are A and N .

The parameters, A and N , can be determined for each
particular stress-strain curve, after Eq. (1) is written
in a different form. This is done in the following manner.
Dividing both sides of Eq. (1) by e gives,

a'/e 8 tanh (A/e)N . I (2)

The inverse of Eq. (2) would then be,
(Ii/o)” It tanh”1 (070) . (3)

from which, I
F

u. = [tanh'l (tr/n] . m)

New dividing both sides of Eq. (M by A gives,
.L

-1 N
l/e = V]! tanh (Vle)
(5)
Taking logarithms of both sides of Eq. (5) results in,

~I
L09 (I/e) = Leg (VA) 4. 75- L09 mt. (We). (6)

Prom.analytic geometry, the equation of a straight
line, in terms of its slepe and y-intercept is: y ==b + mx.
In this equation, y is linear with.x, therefore the equation
' will plot as a straight line. The slope of the straight
line is m, and the y-interdept, b, is the value of y, when
x is equal to zero.

Eq. (6) is also the equation of a straight line, if
the values of e and 0’ are taken from a curve of the equa-
tion, 0’ I e tanh (A/eln. If values of e and 0’ are taken
from.another curve, lying close to a curve of the equation,
0’8 e tanh.(A/e)n, then the plot of log (I/e) against
log tanhfliaye) should approximate a straight line.

Fig. 1 shows the axes chosen for the log tanh71(07e),
and the log (l/e). The values of log tanhI1(07e) are plotted
as the ordinates, with the positive direction extending up-
wards, and the values of log (l/e) as the abscissae, with
the positive direction extending to the right. The axes,
and positive directions, as well as the values at the origin,
were chosen for convenience in constructing and labeling the

axes.

I

C
1—3

a,

[is

«J

 

 

O

 

 

 

4* O

 

 

 

 

 

{\ch':

If the scale of units is the same for both axes in
Pig. 1, the slope of the straight line of Eq. (6), l/N ,
is equal to the tangent of the angle formed by the straight
line with the log tanhfllaye) axis, or any other vertical
line, and is equal to the ratio, b/a . The parameter I in
Eq. (1) is then the reciprocal of l/N or the ratio.a/b in
Pig. 1.

Referring again to Eq. (6), one may note that when the
log tanhIJTOYe) is equal to zero, the log (l/A) is equal to
the log (l/e).’ Therefore in Fig. l, the line representing

zero values of log tanh71(07e) is drawn and labeled I The

0.
projection of the intersection of the straight line plot and

the ID line onto the log (l/e) axis then gives the value of
108 (11") e

Construction:q;gthe Special ngpdinate Papqp

The determination of the parameters in Eq. (1), using
the method developed in the last section, involves finding
the log (l/e) and the log tanh71(07%), for a number of
values of e, and plotting these values of log (I/e) against
the corresponding value of log tanhf}(07e). This requires
a considerable amount of work which can be eliminated by
the construction of a special coordinate paper.

The special coordinate paper is constructed in the

following manner. The log tanhf1(07e) and the log (l/e)

are determined for values of e and O'likely to be encountered

on a stress-strain curve. Values of log tanh71(470) are then
measured along the log tanhI1(07e) axis to a suitable scale.

Lines drawn representing these measured values of log tanh71(°70).
are labeled with the corresponding value of 070. Values of i
log (l/e) are measured along the log (l/e) axis, to the same
scale. Lines representing these mmasured values of log (l/e),

are labeled with the corresponding value of e. A sample of

the special coordinate paper is shown in Fig. 2.

The log tanh71(07e) can now be plotted directly on the
special coordinate paper, from values of“ 07., without cal-
culating log tanh71(07e), and values of log (l/e) can be
plotted directly from values of e without calculating
log (l/e). The parameter A can be read directly from.the
coordinate paper. The parameter N is the tangent of the
angle formed by the straight line with the horizontal axis.

 

10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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11

CHAPTER III

FITTING THE PROPOSED EQUATION TO A TYPICAL CURVE

zippcedure Foljlpweiin the Determination of the Parameters

The preposed equation, 0'8 e tanh (A/e)H , is here fitted
to a typical stress-strain curve, to illustrate the procedure
followed in the determination of the parameters, A and N.

The typical stress-strain curve is shown in Pig. 3.
Table 1 consists of values determined from the curve in
Fig. 3. The first column in Table 1 consists of values of
the unit strain e, chosen at convenient intervals along the
curve, and the corresponding values of stress are shown in
the second colunm.

The values in the third column, 0-: 3/3, are computed
from the values-of stress in the second column, and E.
E, the modulus of elasticity for the given material, is, the
slope of the stress-strain curve of Fig. 3 at the origin.
The values of Cr/e in the fourth column are then computed
from the values of e and 0' in the first and third columns
respectively.

After the calculations in Table l have been completed,
the values of O'/e and e are plotted directly on the special
coordinate paper shown in Fig. 14, with values of 0'/e as the

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TABLE 1

VAflUES DETERMINED FROM TYPICAL STRESS-STRAIN

CURVE, OF FIG. 30 E 3 6.5 x 106 p81.

(T’8 3/2

13

 

9 Stress
1n/1n p31.
0.0005 3,100 o.ooou8 0.955
0.0010 6.300 0.0009? 0.970
0.0015 9.200 0.001u1 0.9h3
0.0020 11.300 0.0017h 0.870
0.0025 13.250 b.0020u 0.816
0.0030 1h,9oo 0.00230 0.765
0.0035 16,300 0.00251 0.71?
o.oouo 17,600 0.00271 0.678
0.00u5 18,800 0.00290 0.6h3
0.0050 19.950 0.00307 0.615
0.0055 20,900 0.00322 0.586
0.0060 21.750 0.00335 0.559
0.0065 22.noo 0.003h5 0.531
0.0070 22.900 0.00353 0.505

 

ordinates and values of e as the abscissae. The parameters,
A and N, can then be determined by drawing the straight line
which gives the best approximation to the points plotted.

The parameter N is the tangent of the angle formed by
the straight line with the horizontal axis, or 0.69. The
parameter A, as shown in Fig. 14., is the projection of the
intersection of the straight line and the Io line onto the
horizontal axis, and is given directly by the values with
which the abscissa is labeled, as 0.0030.

Fig. 5 shows the curve formed when the values of stress,
determined from the equation, 0' 8 e tanh (5/011, with the
parameters A and N determined above, are plotted against the
corresponding values of strain, e. The stress-strain curve
of Pig. ‘3 is superposed on the above curve, and shows that a
good fit is obtained with the values of A and II determined

QbOVOe

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«JZ(IHJ)IMJIJ .H_\JIHJV\J .anI.r3nun___L >Z(LE\JJ H_3.JJU {a .n

 

17

CHAPTER IV

EQUATION (1)1 FITTED To ETHICAL STRESS-STRAIN
CURVES OF DIFFERENT MATERIALS

Typical stress-strain curves, for several engineering
materials, are shown in Figs. 6 to 9. The proposed stress-
strain equation, 0' :- e tanh (PI/flu, has been fitted to each
curve, using the method describedin the last section and
the special coordinate paper shown in Fig. 2.

The parameters, A and N, were determined for each
curve in.Figs. 6 to 9 by drawing the straight line, which
by eye gave the best approximation to the points plotted
on the special coordinate paper. No attempt was made to ob-
tain a better fit to the curves by such means as the method

of least squares. I

1.8

 

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22

CHAPTER V

CONCEUSIONS

This study has resulted in the formulation of the
stress-strain equation, 0-. e tanh (A/e)N, and the
successful fitting of this equation to the actual stress-
strain curves of several metals. In the equation,

(T 8 e tanh (A/e)N, A and N are parameters to be determined
for each stress-strain curve, e is the unit strain and (T'

is equal to the ratio of stress to the modulus of elasticity,
S/E. The modulus of elasticity, E, is the slope of the
actual stress-strain curve at the origin.

The special coordinate paper shown in Fig. 2 was conp
structed to facilitate the determdnation of the parameters,
A and N. In drawing the straight line to represent the
points plotted on the special coordinate paper, better fits
were obtained when precedence was given these points repre-
senting strains in the plastic region.

The stress-strain equation, 0' 8 e tanh (A/e)N, has the
following desirable properties:

(1) The parameters, A and N, can be easily and
accurately determined, for a good fit to experi-
mental data, by using the special coordinate paper
shown in.Pig. 2.

{Q

 

(2)
(3)

(h)

23

The curve passes through the origin.

The slope of the curve at the origin can be
shown to be equal to the modulus of elasticity.
Figs. 6 to 9 show that the equation is applicable

to a wide range of materials.

BIBLIOGRAPHY

Osgood, William.R. Stress—Strain Formulas, Journal of the
Aefiznautical Sciences, vol. 13, pp. ES-hB, (January,
19 .

Eastman, A. J., McDonald, J. C., and Moore, A. A. The Rela-
tion of Stress to Strain in Magnesium.Base Alloys,
ourna of the Aeronautical Sciences, vol. 12, pp. 273-
280, (July, 19h5)e

Klinger, L. J. and Sachs, G. Dependence of the Stress-
. Strain Curves of Cold-werked Meta s pen the Testing
fiirection, Journal of the Aeronautical Sciences, vol.

Franks, R., and Binder, W. O. gbnsioniand Compression
Stress-Strain Characteristics of Cold-Rolled Austenitic
5hromiumpNickel and ChromiumpManganese-Nickelfi Stainless
Steels, Journal of the Aeronautical Sciences, vol. 9,

W. 9438, (September, 19%).

 

 

 

 

 

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