Table Of ContentSTRENGTH AND DEFORMATION IN NONUNIFORM TEMPERATURE FIELDS
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Strength and Deformation
in Nonuniform
Temperature Fields
A collection of scientific papers
edited by
Prof. Ya. B. Fridman
Authorized Translation from the Russian
Springer Science+Business Media, LLC
1964
The original Russian text was published by Atomizdat, the State
Press for Literature in the Field of Atomic Science and Technology,
in Moscow in 1962, for the Moscow Engineering Physics Institute.
Library of Congress Catalog Card Number 63-17641
ISBN 978-1-4757-6606-6 ISBN 978-1-4757-6604-2 (eBook)
DOI 10.1007/978-1-4757-6604-2
© 1964 Springer Science+Business Media New York
Originally published by Consultants Bureau Enterprises, Inc. in 1964.
Reprint ofthe original edition 1964
No part of this publication may be reproduced
in any form without written permission
from the publisher
PREFACE
The often repeated assertion that there is not enough work being done on many problems of mechanical
strength is all the more true ofthermal strength, i.e., the resistance to flow, creep, simple static, impact and
fatigue failure, and the loss of strength brought about by the thermal effects which often accompany mechanical
stresses.
There is no question of the great importance of these problems. Thus it is to be expected that the publi
cation of the present collection will be of interest to investigators in various branches of engineering.
In spite of the rather large number of recent papers dealing with different problems in thermal strength,
there are only a few comprehensive reviews. •
For this reason, a considerable part of the present collection is devoted to a critical review of the results
found in the literature. The presentation of the various papers in the collection is suchthat they can be used in
dependently. The Strengthof Materials Department of the Moscow Engineering Physics Institute extends thanks
in advance to all the organizations and individuals who will take the trouble to communicate (address: Strength
of Materials Department, Moscow Engineering Physics Institute, Kirov Street, 21, Moscow) their suggestions and
critical comments on the material published in the present collection.
• Among such reviews we can mention the book by B. Gatewood, and the papers by s. V. Serensen and P. I.
Kotov, A. Freudenthal, Manson, et al.
CONTENTS
Some of the Laws Governing Mechanical and Thermal Strength
by Ya. B. Fridman •...................... 2
Thermal Stresses and Their Calculation
by E. M. Morozov and Ya. B Fridman ..... 17
Thermal Fatigue and Thermal Shock
by N. D. Sobolev and V. I. Egorov ..... . 62
Fundamentals of Creep Calculations on Nonuniformly Heated Parts
by B. F. Shorr •........................... 116
Thermal Stability of Plates and Shells
L. A. Shapovalov. ......... . 159
SOME OF THE LAWS GOVERNING
MECHANICAL AND THERMAL STRENGTH
Ya. B. Fridman
Until recently, effectsof mechanicalloads (such as weight), inertia, the pressure of solids, liquids, gases,
etc., have received most of the attention in studies of strength of materials and structural elements.
At the present time, there are many branches of science and engineering in which very high operating
temperatures are used. At the sametime the parts required have become more complex. The nonuniform tem
perature fields that are encountered on going from one part of a body to another (at the same instant of time)
and in every part of the body (as a function of time) can often lead to failures, with very smallloadings and
even with no mechanicalloads at all.
In the broad sense of the word, failures of mechanical or thermal strength may be considered to include
the following:
1. Reduction of the bearing strength of a structural element (for example, in failure), loss of stability, etc.
2. Failure of the element to operate normally (even while retaining its strength) as a result of excessive
deformation, or the development of a failure (for example, deformation beyond the permissible Iimit), or loss
of air-tightness when flaws occur, etc.
In this sense, one may speak of thermal strength or of analyzing thermal stresses, i.e., flow, creep, simple
static, fatigue and impact:failure, and loss of strength resulting from purely thermal effects. There are various
physical fields which Iead to nonuniform deformations, which could serve as a basis for treatment of other types
of strength - e.g., magnetic or electrical.
Since, in almost all solids the temperature effect produces thermal expansion (or compression), which in
the majority of practical cases is restrained, • it is obvious that the stresses and deformations from temperature
effects differ from those due to externalloads only in the nature of the source.
Since thermal stresses a.re always determined by the deformation and not by the stresses, they relax to
some extent with increase in deformation (or motion), while mechanicalloadings may be either strongly relax
ing or completely nonrelaxing, for example, in a suspended load (Fig. 1).
A comparison of mechanical and thermal effects is shown for several cases of loss of strengthin Table 1.
If it is assumed that an elastic system under thermal Stresses is conservative, and that the forces are equal
to the derivatives of the potential energy, then for any temperature field, certain surface and volume forces may
be calculated the effects of which are equivalent to the effect of the given thermal field. In other words, if a
given thermal field is acting, loss of strength can occur at certain critical temperatures, and the critical values
of the externalloads may be calculated for any given state. Here, a uniform temperature field corresponds to
• Thermal expansion of solids is prevented by adjacent zones at another temperature, adjacent parts, or (even in
a uniform temperature field) nonuniform or anisotropic properdes at various points in the solid.
1
~echanical loading Thermal the effect of surface loads alone, while a nonuniform
loading
A temperature field corresponds to the effect of both sur
face (Ps) and volume (Pv) loads. In this sense, an analogy
~
may be drawn between purely thermal and purely me
chanical phenomena. At the same time, however, there
are fundamental differences between the effects of me
chanical and thermalloadings.
Deformation complete failure (separation) of the patt seldom
Fig. 1. Relaxation of mechanical and thermal occurs under purely thermalloading. Typical ofthermal
Stresses: (A) nonrelaxing mechanicalloading (very failure (patticularly under multiply repeated loadings) is
large elastic energy reserve, case of fixed stress); •saturation" of the patt with flaws, which is accounted
(B) weakly relaxing; (C) strongly relaxing mecha for by both the local nature and the rapid relaxation of
nicalloading (elastic energy supply very sman). the thermal stresses. Thus, the development of flaws
Thermal stresses are, as a rule, strongly relaxing. (which under mechanicalloading with a given force
would increase the stresses and hence aceeierate the pro
cess), with even sman amounts of motion, lowers the thermal stresses and keeps the flaws from propagating, so
that they do not get an the way through the cross section of the patt. lf a thermalload is applied again, the
largest thermal stresses occur at other places (since the flaws that have already been formed reduce the amount
of local coupling, making local thermal deformation easier, and, thus, •uruoad• the zones in the immediate
vicinity). Thus, whenever thermalloadings are repeated, the flaws are always formed in some new region ofthe
part.
In the present paper, • a shott discussion and comparison is given of some of the laws governing mechani
cal and thermal strength. The discussion is given mainly for the laws governing macroscopic processes, and
these may be substantiany different from those holding microscopicany. For example, microscopic deformation
and failure are often discontinuous in nature, although they usuany seem tobe continuous on a macroscopic
scale.
Various Cases of Loading
In view of the great variety of loadings that occur in tests and the even greater variety encountered under
operating conditions, we shan consider the possible classifications of the fundamental types of loading. First of
an we must distinguish between unsustained and sustained systems. t
In the first case, the loaded part (sample or assembly) transmits from a source with a definite supply of
energy, some given initial force or displacement, and then plastic deformation or failure develops with time •
.E xamples of this among mechanicalloadings eire (1) a holt, after it has been drawn up, if no additional forces
are applied to it, and (2) a tank after it has been given a constant internalliquid or gas pressure. Among ther
malloadings, it can be seen in the production of a sustained (steady-state) temperature difference, constant in
time, for example, between the inside and outside surfaces of a tube or vessel.
In sustained systems, the part receives the load P or the displacement ß, not simply at the statt, but dur
ing the rest of the deformation, either by increase in P or ß, or by continuous repetition. Examples of sustained
systems are provided by the various cases of mechanical and thermal fatigue, as wen as by static tests under in
crease in load (or increase in temperature gradient). Both sustained and unsustained systems can show an four
of the kinetic periods of deformation and failure given below. Of course, in sustained systems, the course of the
process is determined by both the initialload and resistance, and by the nature of the energy "feed• under load.
The distinction between sustained and unsustained systems may change in the course of loading. For example,
if the end of failure is passed through very quickly, the sample may become disconnected from the source of
loading, and become unsustained. On the other hand, in an initiany unsustained system, for example, after
• This paper makes use of the results of work done by the author in cooperation with T. K. Zilova. with B. A.
Drozdovskii [1,2] and others, as wen as of a nurober of the papers cited in the references.
tUnsustained and sustained systems correspond to free and forc~d oscillations, respectively.
2
T ABLE 1. Various Cases of Loss of Strength
Under Mechanical and Thermal Loading
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!.".i- tcr=f(o.,E, Pncr" f< E,h,
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h,L,m,R) L, m)
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V
Uniform heatlng
,T '
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-Ei (to)= f(a,E (No)cf f(E,
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formation of a neckdown or a flaw, the system may become sustained by a narrow region being continuously fed
with energy from other parts of the body as it becomes isolated.
Another important distinction may be drawn between methods of loading, on the basis of the way in which
the load is applied to the part: there may be a fixed Ioad, i.e., one which does not relax during deformation,
or the displacement may be fixed, or there may be mixed forms. Examples are given in Table 2 from which
it may be seen that it is typical of temperature loading to have fixed displacement (since what the temperature
gradient determines is the displacement). The importance of this fact is shown in the paper by N. D. Sobolev
and V. I. Egorov, •Thermal fatigue and thermal shock" (see the present collection, page 62).
Four basic cases should be distinguished for either thermal or mechanical effects, depending on the nature
of the loading:
1. Mechanical orthermal shock, in which wave processes begin to play a substantial role (e.g., in sudden
local bathing of a part of considerable length by metallic coolant), as do inertial forces (e.g., even in light
metals the effect of a shock velocity of several hundred meters per second is determined mainly by the increase
in inertial resistance, rather than by change in mechanical properties). Under static Ioads, neither wave nor in
ertial corrections play any essential role. This is the specific feature of shock loading which does not show up
under static load. Note that many mechanical properties usually designated by the word shock (or impact), are
not of this nature at all- e.g., the "impact toughness" when a notched sample is bent with a swinging pendulum
3
e Stresses of the First and Second Kinds [10] and microtemperature stresses Microtemperature stresses (of the second kind) occur, especially when the lattice has low symmetry, forexample, in cadmium, tin, etc., as a result of anisotropy and differ ence in thermal expansion of the elements in the structure. Relaxation of microstresses, because of their large extent and dispersion, goes comparatively slowly when kept at ele vated temperatures. Large effect, since the structural inhomogeneity which to act even when the produces the microstresses continues is at a constant temperature. solid Small effect, since microrelaxation processes usually can not occur to any appreciable extent when the temperature is changed. No appreciable effect, since the microstresses are deter mined mainly by local conditions. Increasing the grain size reduces the number of structural elements, and thus greatly reduces the residual deformation when the temperature change is repeated. Large effect, sometimes changes the growth coefficient in temperature changes by a factor of 20.
r
u -
amparisan of the Effects of Various Factars on Temperat Special features of macro Macrotemperature stresses (of the first kind) occur as a result of temperature difference between differ ent zones in the solid. The surface zones are usual ly more strongly stressed than the interior zones, and thus the dimensions of the surface zones are less than those of the interior zones. As a result, in rapid heat ing, the total deformation of the solid is considerably less (the external hot zones are small), than in rapid cooling (the extent of the interior hot zones large). is Little effect, since (with the exception of a steady state temperature gradient) stresses only occur when the temperature is changed. some Large effect, determining the magnitude and times the sign of the stresses. Small effect, particularly with increase in the sur face-to-volume ratio (for the same value of cross section). Little effect. Often no difference between single and polycrystals. Small effect.
C
2.
T ABLE pe of factor tem he upper of the cycle n heating and ates d absolute ns of the solid e
Ty Kept at tperature Change icooling r Shape andimensio Grain siz Texture
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