Described are the mathematical model and solution methodologies for analyzing the structural response of thin, metallic elasto-viscoplastic shell structures under large thermomechanical loads and their non-isothermal buckling behavior. Among the system responses associated with these loads and conditions are snap-through, buckling, thermal buckling, and creep buckling. This geometric and material nonlinearities (of high order) can be anticipated and are considered in the model and the numerical treatment

Riff, Richard

Simitses, G. J.

English

NASA Technical Reports Server

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Non-Isothermal Buckling Behavior Of
Viscoplastic Shell Structures
by
R. Riff I and G. J Simitses 2
Georgia Institute of Technology
Atlanta, Georgia 30332
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ABSTRACT
The paper describes the mathematical model and solution
methodologies for analyzing the structural response of thin,
\ metallic elasto-viscoplastic shell structures under large ther-
momechanical loads. Among the system responses associated with
these loads and conditions are snap-through, buckling, thermal
buckling and creep buckling. Thus geometric and material non-
linearities (of high order) can be anticipated and are considered
in the model and the numerical treatment.
The problem of inelastic analysis of shell structures has been
investigated recently by Kojic' and Bathe I. They used the "effe-
ctive-stress-function" algorithm to compute plastic and creep
effects on the behavior of shell like structures. The effects of
inelastic material behavior on stability of shells found their way
into the literature since the late 1970's. The paper by Miyazaki
and Ando 2 deals with creep buckling of perfect spherical shells
subjected to pressure loading and considers only the effects of
steady-state creep. Xirochakis and Jones 3 have studied axisym-
metric and bifurcation creep buckling of externally pressurized
spherical shells under the condition of secondary creep only.
Botros and Bienek 4 presented a numerical treatment of the creep
buckling of these configurations. Their work includes the effects
of elastic strain, primary and secondary creep strains and creep
recovery. The influence of temperature and viscous effects on
dynamic buckling of shells has been considered by Wojewodzki and
Bukowski 5'6. The book by Owen and Hinton 7 gives a list of references
for the applications of finite element methods to the problem of
creep buckling of structures.
As far as the authors know no work has been reported on the
non-isothermal buckling behavior of elasto-viscoplastic shell
structures.
The prediction of inelastic behavior of metallic materials at
elevated temperatures has increased in importance in recent years.
The operating conditions within the hot section of a rocket motor
or a modern has turbine engine present an extremely harsh thermo-
IAssistant Professor of Aerospace Engineering, Member AIAA
2Professor of Aerospace Engineering, Associate Fellow of AIAA,
Member ASME
Structures
https://ntrs.nasa.gov/search.jsp?R=19890003560 2020-03-20T04:34:33+00:00Z
mechanical environment. Large thermal transients are induced each
time the engine is started or shut down. Additional thermal
transient from an elevated ambient, occur whenever the engine power
level is adjusted to meet flight requirements. The structural
elements employed to construct such hot sections, as well as any
engine components located therein, must be capable of withstanding
such extreme conditions. Failure of a component would, due to the
critical nature of the hot section, lead to and immediate and
catastrophic loss in power and thus cannot be tolerated. Conse-
quently, assuring satisfactory long term performance for such
components is a major concern for the designer.
In previous papers 8'9'I° the authors have presented a con-
stitutive law for thermo-elasto-viscoplastic behavior of metallic
materials, in which the main features are: (a) unconstrained strain
and deformation kinematics, (b) selection of reference space and
configuration for the stress tensor, bearing in mind the rheologies
of real materials, (c) an intrinsic relation which satisfies
material objectivity, (d) thermodynamic consistency, and (e) proper
choice of external and internal thermodynamic variables.
The problems of buckling of shallow arches under static
thermo-mechanical loads was investigated in Ref. i0. It was shown
there that the material constitutive equation are capable of
capturing all non-isothermal, elasto-viscoplastic characteristics.
Furthermore, the method is capable of predicting response which
includes pre and postbuckling behavior due creep and plastic
effects.
The non-isothermal inelastic analysis of shell structures
represent some major challenges. In order to perform an effective
and accurate analysis, efficient formulation and solution stra-
tegies must be employed.
A complete true ab-inito rate theory of kinematics and
kinetics for curved thin structures, without any restriction on
the magnitude of the strains or the deformation, was formulated.
The time dependence and large strain behavior are incorporated
threugh the introduction of the time rates of the metric and
curvature in two coordinate systems; a fixed (spatial) one and a
convected (material) coordinate system. The relations between the
time derivative and the convariant derivatives (gradients) have
been developed for curved space and motion, so that the velocity
components supply the connection between the equations of motion
and the time rate of change of the metric and curvature tensors.
A time and temperature dependent elasto-viscoplastic model
was formulated s1° in this convected material system to account for
finite strains and rotations. The history and temperature
dependence were incorporated through the introduction of internal
variables. The choice of these variables, as well as their
evolution, was motivated by phenomenological thermodynamic
considerations 8.
The obtained complete rate field equations consist of the
principles of the rate of the virtual power and the rate of
conservation of energy, of the constitutive relations, and of
boundary and initial conditions. These equations provide a sound
basis for the formulation of the adopted finite element solution
procedures.
The derived shell theory, in the least restricted form, before
any simplifying assumptions are imposed, has the following
desirable features:
(a) The two-dimensional, impulse-integral form of the equations
of motions and the Second Law of Thermodynamics (Clausius-
Duhem inequality) for a shell follow naturally and exactly
from their three-dimensional counterparts.
(b) Unique and concrete definitions of shell variables such as
stress resultants and couples, rate of deformation, spin and
entropy resultants can be obtained in ter_s of weighted
integrals of the three-dimensional quantities through the
thickness.
(c) There are n_oo series expansions in the thickness direction.
(d) There is no need for making use of the Kirchhoff Hypotheses
in the kinematics.
(e) All approximations can be postponed until the descretization
process of the integral forms of the First Law of Thermo-
dynamics.
A by-product of the descent from three-dimensional theory is
that the two-dimensional temperature field (that emerges) is
not a through-the-thickness average, but a true point by point
distribution. This is contrary to what one finds in the
literature concerning thermal stresses in the shell.
(f)
To develop geometrically nonlinear, doubly curved finite shell
elements the basic equations of nonlinear shell theories have to
be transferred into the finite element model. As these equations
in general are written in tensor notation, their implementation
into the finite element matrix formulation requires considerable
effort. The nonlinear element matrices are directly derived from
the incrementally formulated nonlinear shell equations, by using
a tensor-oriented procedure. For this formulation, we are using
the "natural" degrees of freedom per mid-surface shell node: three
incremental velocities and the rates of rotations about the
material coordinates in a mixed form.
The quasi-linear nature of the principle of the rate of
virtual power suggests the adoption of an incremental approach to
numerical integration with respect to time. The availability of
the field formulation provides assurance of the completeness of
the incremental equations and allows the use of any convenient
procedure for spatial integration over the domain V. In the
present instance, the choice has been made in favor of a simple
expansion in time for the construction of incremental solutions
from the results of finite element spatial integration of the
governing equations.
The procedure employed permits the rates of the field
formulation to be interpreted as increments in the numerical
solution. This is particularly convenient for the construction of
incremental boundary condition histories.
Finite element solution of any boundary-value problem involves
the solution of the equilibrium equations (global) together with
the constitutive equations (local). Both sets of equations are
solved simultaneously in a step by step manner. The incremental
form of the global and local equations can be achieved by taking
the integration over the incremental time step _t=t÷.-t- TheJl .J"
rectangular rule has been applied to execute the resultlng time
integration.
Clearly, the numerical solution involves iteration. A
simplified version of the Riks-Wempner constant-arc-length method
has been utilized. This iteration procedure which is a genera-
lization of the displacement control method also allows to trace
the non-linear response beyond bifurcation points. In contrast to
the conventional Newton-Raphson techniques, the iteration of the
method takes place in the velocity and load rate space. The load
step of the first solution in each increment is limited by
controlling the length ds of the tangent. Either the length is
kept constant in each step or it is adapted to the characteristics
of the solution. In each step the triangular-size stiffness matrix
has to be checked for negative diagonal terms, indicating that a
critical point is reached.
Several examples of shell cylindrical panels and shallow
spherical caps of different materials subjected to different
loading and temperature conditions have been analyzed. As an
example of the results the central deflection time history and the
influence of temperature change of a thin, imperfect, cylindrical
shell panel made of carbon steel C-45 is shown in the figure shown
herein. The panel is simply supported on all sides, and subjected
to inplane loads along the generator. The applied load of 20
ibs/in, is well below the linear critical (buckling) load for this
geometry, which is 42.15 ibs/in. At a temperature of 50 ° F the
shell is in a primary creep state for the first 28 minutes,
reaching a deflection of 0.2 in. and the critical time for creep
buckling (this implies that the deflection becomes unbounded) is
35 minutes. At temperature of 500 ° F the shell 'maps' into its
post-buckled configuration almost immediately but the critical time
for creep buckling remains almost unchanged. The dashed line in the
figure represents a non-isothermal process where the temperature
was suddenly increased from 50 ° F to 500 ° F after 0.3 hrs. As a
result the shell snaps-through to its post buckled position at 500 °
F with small over-shoot and reaches its critical time of creep
buckling two minutes sooner (33 min.).
The study shows that in the presence of high temperature and
viscoplasticity, the process of shell buckling assume an entirely
new character. While the stability phenomena still exist under
sufficiently large loads, buckling develops, as a time and tempera-
ture dependent deformation process under constant or variable
thermomechanical loads of magnitude smaller than the purely elastic
critical values. If the elastic behavior of a structure displays
limit points and snap-through phenomena, the deformation process
of creep buckling become even more complicated and it usually
exhibits a combination of snapping and creep responses.
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CYLINDER PANEL CREEP RESPONSE
Acknowledgement
The work is sponsored by the NASA-Lewis Research Center under
NASA Grant No. NAG3-534. The program manager is Dr. Chris Chamis.
The financial support provided by NASA is gratefully acknowledged.
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References
M. Kojic and K. J. Bathe, "Thermo-Elastic-Plastic and Creep
Analysis of Shell Structures", Computer and Structures, Vol.
26, No. 1/2, 1987, pp. 135-143•
G. Miyazaki and Y. Ando, "Parametric Analysis of Creep
Buckling of a Shallow Spherical Shell using the Finite Element
Method", Nuclear Enqineerinq and Design, 1977.
P. C. Xirochakis and N. Jones, "Axisymmetric and Bifuraca-
tion Creep Buckling of Externally Pressurized Spherical
Shells", Solids and Structures, Vol. 16, 1980, pp. 131-148.
F. R. Botros and M. P. Bienek, "Creep Buckling of Struc-
tures", Proceedings of the AIAA/ASME/ASCE/AHS 24th SDM Oonfe-
rence, AIAA Paper No. 83-0864, May 1983.
W. Wojewodzki and R. Bukowski, "The Influence of the Yield
Function Nonlinearity and Temperature in Dynamic Buckling of
Viscoplastic Cylindrical Shells", Applied Mechanics, Vol. 51,
1984, pp. 114-120•
R. Bukowski and W. Wojewodzki, "Dynamic Buckling of Visco-
plastic Spherical Shell", Solids and Structures , Vol. 20,
No. 8,1984, pp. 761-776.
D. R. J. Owen, and E. Hinton, Finite Elements in Plasticity,
Pineridge Press, Swansea, U.K., 1980.
R. Riff, R. L. Carlson, and G. J. Simitses., "Thermody
lly Consistent Constitutive Equations for Noniso t h e r m a 1
Large -Strain, Elastoplastic, Creep Behavior," AIAA, Vol,
25, 1987, pp. 114-122.
R. Riff and G. J. Simitses, ,,Thermo-Elasto-Viscoplastic
Analysis of Problems in Extension And Shear", Computer &
Structures, Vol. 29, No. 2, 1988, pp. 293-300.
G. J. Simitses and R. Riff., "Non-Isothermal Elasto Visco-
plastic Snap-Through and Creep Buckling of Shallow Arches"
Proceedings of the 28th Structures, Structural Dynamics and
Material Conference, Montrey, California, April 1987, p p •
466-472•


Non-isothermal buckling behavior of viscoplastic shell structures

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