Toward a Methodology for Studying Large Plastic Deformations under Simple Loading
Abstract Generated abstract
The paper addresses the problem of maintaining simple loading during experiments on large plastic deformation, where conventional stepwise correction after deformation measurements can violate the required constancy of the principal stress ratio. It develops loading programs for thin walled tubular specimens subjected to combined axial force and internal pressure, using true stresses, logarithmic strains, incompressibility of plastic deformation, and assumed single flow curves in several coordinate systems. Formulas are derived for the necessary pressure and axial force over different ranges of the principal stress ratio, with an example calculation for copper. The authors conclude that deviations from simple loading grow with plastic strain, that true stress based programs are needed for certain stress ratio ranges, and that the proposed programs allow continuous loading and recording of deformation characteristics along the loading path.
Full Text
On the Methodology for Studying Large Plastic Deformations under Simple Loading
K. K. Pakhotin, L. M. Sedokov
(Presented by the city scientific conference of young scientists and specialists)
In studying large plastic deformations it is necessary to distinguish between simple and complex loading. By simple loading of a body is meant such loading under which the directing tensors of the stresses at each point of the body retain values constant for that point [1]; consequently, in the case of studying a plane stress state it is necessary and sufficient that the ratio between the principal stresses, not equal to zero, remain constant during the loading process up to fracture.
Such a requirement can be fulfilled if the magnitude and ratio of the principal deformations at the points of the loaded body are taken into account.
Usually [2] the experiment is conducted as follows: after specifying the ratio of the principal stresses, the specimen is loaded; then, after measuring the deformation, the actual true stresses acting in the specimen at the given moment are calculated. Such a calculation reveals a violation of the requirements of simple loading and makes it possible to carry out the necessary correction of the loading program.
Such a methodology for studying large plastic deformations substantially complicates the experiment and, most importantly, does not ensure simple loading within each stage.
Therefore the need arose to develop a special loading program taking into account the hardenability of the material.
Let us make the following assumptions: 1. No change in volume occurs during plastic deformation. Hence the coefficient of logarithmic transverse deformation is equal to 0.5. 2. Elastic deformations are negligibly small; therefore the relations between the principal deformations and the principal true stresses are taken in the form [1]
\[ e_1:e_2:e_3=(s_1-0.5\,s_2):(s_2-0.5\,s_1):[-0.5(s_1+s_2)]. \tag{1} \]
In addition, for an accurate accounting of the behavior of the material in the region of large plastic deformations, a definite decision is made regarding the existence of a single flow curve, which is an inevitable difficulty in calculating a simple-loading program. Three main types of single flow curves are known in the coordinates:
a) maximum shear stresses — maximum logarithmic shear \((t-\tg)\);
b) intensity of normal stresses — intensity of deformation \((s_i-e)\);
c) the specific work of plastic deformation—the maximum logarithmic shear \((a—g)\) [3].
As special investigations on steel, copper, and brass show, the unified \(a—g\) curve is satisfactorily fulfilled throughout the entire deformation process up to fracture (Fig. 1). The subsequent calculations will be carried out on the basis of this flow curve.
Fig. 1. Flow curves. According to the authors’ experiments
In testing thin-walled tubes under the action of a combined load \((P, q)\), the following average stresses arise in their walls [4]:
\[ \begin{aligned} s_l &= \frac{P+qF}{f} - \frac{q}{2},\\ s_t &= \frac{2qF}{f} - \frac{q}{2},\\ s_r &= -\frac{q}{2} \end{aligned} \left\} \right., \tag{2} \]
where \(P\) and \(q\) are the axial force and internal pressure,
\[
F=\frac{\pi}{4}\left(\frac{D+d}{2}\right)^2
\]
is the area of the circle of mean radius,
\[
f=\frac{\pi}{4}(D^2-d^2)
\]
is the cross-sectional area of the tube,
\(D\) and \(d\) are the current outer and inner diameters of the tube,
\(s_l\) is the axial true stress,
\(s_t\) is the circumferential true stress,
\(s_r\) is the radial true stress.
radial stress, as well as the second terms in the first and second lines of (2), will henceforth be neglected, in view of their obvious smallness; then the stress state in the tube walls may be regarded as plane.
In the process of deformation the cross section of the tube changes so that, taking into account the true deformation characteristics, its current value is expressed as
\[ F=F_0 \exp\{2e_t\}, \qquad f=f_0 \exp\{e_t+e_r\}, \tag{3}\tag{4} \]
where \(F_0\) and \(f_0\) are the initial values for \(F\) and \(f\),
\[ e_t \text{ is the circumferential strain,} \]
\[ e_r \text{ is the radial strain.} \]
Taking, for simple loading,
\[ n=s_1:s_2, \tag{5} \]
from (2), with (3) taken into account, we obtain
\[ \frac{P}{q}=F_0(2n-1)\exp\{2e_t\}. \tag{6} \]
Let us introduce certain characteristics of the form of the deformed state [5], [6]:
\[ \kappa=\frac{e_3}{e_1}, \tag{7} \]
\[ \beta=\frac{2}{3}\left(2+\frac{e_2}{e_{1,3}}\right). \tag{8} \]
It may now be written that
\[ g=e_1(1-\kappa). \tag{9} \]
On the basis of the adopted single curve c), the expression for the maximum shear stress in the general case of the deformed state is obtained as [5]
\[ t=\frac{9}{8}\beta\frac{da}{dg}=c\beta. \tag{10} \]
Let us consider the stress-strain state for the ratio \(-1<n<1\), assuming at the same time a plasticity condition that has two planes of symmetry in the space of principal stresses [7].
Depending on the ratio of the principal strains \(\kappa\), three cases must be considered in calculating the program:
1st case. \(0.5<n<1;\ e_1=e_t;\ e_2=e_\theta;\ e_3=e_r;\)
\[ \kappa=-\frac{1+n}{2-n}, \tag{11} \]
\[ \beta=\frac{2}{n+1}, \tag{12} \]
\[ t=\frac{s_t}{2}, \tag{13} \]
or, taking (2), (4), (9), and (10) into account, we rewrite
\[ t=q\frac{F_0}{f_0}\exp\{g\}-\frac{2}{n+1}c. \]
Hence
\[ q=\frac{2}{n+1}\cdot \frac{f_0}{F_0}\,c\cdot \exp\{-g\}. \tag{14} \]
Using (9) and (14), from (6) we obtain
\[ P=\frac{2(2n-1)}{n+1}\,f_0\cdot c\cdot \exp\left\{\frac{1-2n}{3}\,g\right\}. \tag{15} \]
II case. \(0<n<0.5;\ e_1=e_t;\ e_2=e_i;\ e_3=e_r.\)
In this case
\[ \beta=\frac{2}{2-n}. \tag{16} \]
Then the internal pressure and axial force are determined as
\[ q=\frac{2}{2-n}\cdot \frac{f_0}{F_0}\,c\cdot \exp\{-g\}, \tag{17} \]
\[ P=\frac{2(2n-1)}{2-n}\,f_0\cdot c\, \exp\left\{\frac{1-2n}{3}\,g\right\}. \tag{18} \]
III case. \(-1<n<0;\ e_1=e_t;\ e_3=e_i;\ e_2=e_r.\)
Here
\[ \kappa=\frac{2n-1}{2-n}, \tag{19} \]
\[ \beta=\frac{2(1-n)}{2-n}, \tag{20} \]
\[ t=\frac{s_t}{2}(1-n). \tag{21} \]
Now for \(q\) and \(P\) we find the following expressions:
\[ q=\frac{2}{2-n}\cdot \frac{f_0}{F_0}\,c\cdot \exp\left\{+\frac{1}{1-n}\,g\right\}, \tag{22} \]
\[ P=\frac{2(2n-1)}{2-n}\,f_0\cdot c\, \exp\left\{\frac{1-2n}{3(1-n)}\,g\right\}. \tag{23} \]
Taking the flow curve in the coordinates \(s_i-e_i\), for all the considered ratios of the principal stresses \((-1<n<1)\), for \(q\) and \(P\) we obtain
\[ q=s_i\cdot m\frac{f_0}{F_0}\exp\{-3e_i m\}, \tag{24} \]
\[ P=s_i\cdot m\cdot f_0(2n-1)\exp\{(1-2n)e_i m\}, \tag{25} \]
\[ m=[4(1-n+n^2)]^{-1/2}. \tag{26} \]
If the material corresponds to a single flow curve \(t-g\), the program takes the form:
I case. \(-1<n<0.\)
\[ q=t\,\frac{1}{(1-n)}\,\frac{f_0}{F_0}\, \exp\left\{-\frac{1}{1-n}\,g\right\}, \tag{27} \]
\[ P=t\,\frac{2n-1}{1-n}\,f_0\, \exp\left\{\frac{1-2n}{3(1-n)}\,g\right\}. \tag{28} \]
II case. \(0<n<1.\)
\[ q=t\,\frac{f_0}{F_0}\exp\{-g\}. \tag{29} \]
Fig. 2. Loading program for copper at
\(n = 1.0\) and \(n = 0.4\)
Fig. 3. Deviations from simple loading in the case of calculating the program by nominal stresses at \(e_i = 5, 10, 20\%\)
\[ P=t(2n-1)f_0\exp\left\{\frac{1-2n}{3}g\right\}. \tag{30} \]
The loading program for copper, calculated under the assumption of the existence of a single \(a—g\) curve, is presented in Fig. 2. In the calculation it was assumed: \(D=30\) mm, \(d=28\) mm.
The absolute deviation obtained in calculating the simple-loading program from the nominal stresses \((\sigma_1,\sigma_2)\) is determined from the following equality:
\[ \Delta=n_0-n=\frac{(2n-1)(\exp\{2e_t\}-1)}{2}, \tag{31} \]
where \(n_0=\sigma_1/\sigma_2\). Dependence (31) is presented in Fig. 3.
Conclusions
-
With increasing plastic deformation, the violation of the requirement of simple loading increases.
-
For \(n\ge 0.8\) and \(n\le 0.2\), the program should be calculated from the true stresses. The program is calculated under the existence of a single flow curve in the coordinates:
\(a—g\) — by formulas (14, 15, 17, 18, 22, 23),
\(s_t—e_t\) — by formulas (24, 25),
\(t—g\) — by formulas (27, 28, 29, 30).
-
For \(0.2<n<0.8\), the program may be calculated from the nominal stresses.
-
Owing to the obtained program, the process of loading the specimen becomes continuous, and it becomes possible to record the deformation characteristics over the entire loading path and at any point of it by means of an oscillograph.
References
- S. D. Ponomarev et al. Strength Calculations in Mechanical Engineering. Vol. II, Moscow, 1958.
- A. M. Zhukov. Engineering Collection. Vol. XX, Moscow, 1954.
- L. M. Sedokov. Resistance to Large Plastic Deformations. Tomsk, 1956.
- V. V. Lode. In: Theory of Plasticity. Moscow, 1948.
- V. G. Osipov. In: Problems of Metallurgy. Moscow, 1953.
- P. O. Pashkov. Plasticity and Fracture of Solids. Moscow, 1949.
- D. D. Ivlev. Applied Mathematics and Mechanics. Vol. XXII, Moscow, 1958.