Modern experimental stress analysis: completing the solution of partially specified problems

435 Pages·2004·6.831 MB·English

by  James F. Doyle

MODERN EXPERIMENTAL
STRESS ANALYSIS
completing the solution of partially specified problems
James F. Doyle
Purdue University, Lafayette, USA
MODERN EXPERIMENTAL
STRESS ANALYSIS
MODERN EXPERIMENTAL
STRESS ANALYSIS
completing the solution of partially specified problems
James F. Doyle
Purdue University, Lafayette, USA
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Contents
Preface ix
Notation xi
Introduction 1
1 Finite Element Methods 11
1.1 Deformation and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Tractions and Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Governing Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Material Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.6 Some Finite Element Discretizations. . . . . . . . . . . . . . . . . . . . . 43
1.7 Dynamic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.8 Geometrically Nonlinear Problems. . . . . . . . . . . . . . . . . . . . . . 67
1.9 Nonlinear Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2 Experimental Methods 83
2.1 Electrical Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.2 Digital Recording and Manipulation of Signals . . . . . . . . . . . . . . . 90
2.3 Electrical Resistance Strain Gages . . . . . . . . . . . . . . . . . . . . . . 101
2.4 Strain Gage Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5 Motion and Force Transducers . . . . . . . . . . . . . . . . . . . . . . . . 116
2.6 Digital Recording and Analysis of Images . . . . . . . . . . . . . . . . . 122
2.7 Moire´ Analysis of Displacement . . . . . . . . . . . . . . . . . . . . . . . 138
2.8 Holographic Interferometry. . . . . . . . . . . . . . . . . . . . . . . . . . 146
2.9 Photoelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3 Inverse Methods 171
3.1 Analysis of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . 172
3.2 Parametric Modeling of Data. . . . . . . . . . . . . . . . . . . . . . . . . 178
3.3 Parameter Identification with Extrapolation . . . . . . . . . . . . . . . . . 190
3.4 Identification of Implicit Parameters . . . . . . . . . . . . . . . . . . . . . 196
3.5 Inverse Theory for Ill-Conditioned Problems . . . . . . . . . . . . . . . . 200
3.6 Some Regularization Forms . . . . . . . . . . . . . . . . . . . . . . . . . 207
viii Contents
3.7 Relocation of Data onto a Grid Pattern . . . . . . . . . . . . . . . . . . . 212
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4 Static Problems 219
4.1 Force Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . 220
4.2 Whole-Field Displacement Data . . . . . . . . . . . . . . . . . . . . . . . 229
4.3 Strain Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
4.4 Traction Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
4.5 Nonlinear Data Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4.6 Parameter Identification Problems . . . . . . . . . . . . . . . . . . . . . . 254
4.7 Choosing the Parameterization . . . . . . . . . . . . . . . . . . . . . . . . 265
4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
5 Transient Problems with Time Data 277
5.1 The Essential Difficulty . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.2 Deconvolution using Sensitivity Responses . . . . . . . . . . . . . . . . . 280
5.3 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
5.4 Scalability Issues: Recursive Formulation . . . . . . . . . . . . . . . . . . 295
5.5 The One-Sided Hopkinson Bar . . . . . . . . . . . . . . . . . . . . . . . 302
5.6 Identifying Localized Stiffness and Mass . . . . . . . . . . . . . . . . . . 306
5.7 Implicit Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . 313
5.8 Force Location Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
6 Transient Problems with Space Data 331
6.1 Space–Time Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . 332
6.2 Preliminary Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
6.3 Traction Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
6.4 Dynamic Photoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
6.5 Identification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
6.6 Force Location for a Shell Segment . . . . . . . . . . . . . . . . . . . . . 360
6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
7 Nonlinear Problems 363
7.1 Static Inverse Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
7.2 Nonlinear Structural Dynamics. . . . . . . . . . . . . . . . . . . . . . . . 371
7.3 Nonlinear Elastic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7.4 Elastic-Plastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
7.5 Nonlinear Parameter Identification . . . . . . . . . . . . . . . . . . . . . . 386
7.6 Dynamics of Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
7.7 Highly Instrumented Structures . . . . . . . . . . . . . . . . . . . . . . . 399
7.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Afterword 411
References 413
Index 425
Preface
Thisbookisbasedontheassertionthat,inmodernstressanalysis,constructingthemodel
isconstructingthesolution—thatthemodelisthesolution.Butallmodelrepresentations
of real structures must be incomplete; after all, we cannot be completely aware of every
material property, every aspect of the loading, and every condition of the environment,
for any particular structure. Therefore, as a corollary to the assertion, we posit that a
very important role of modern experimental stress analysis is to aid in completing the
construction of the model.
What has brought us to this point? On the one hand, there is the phenomenal growth
of finite element methods (FEM); because of the quality and versatility of the commer-
cial packages, it seems as though all analyses are now done with FEM. In companies
doing product development and in engineering schools, there has been a corresponding
diminishing of experimental methods and experimental stress analysis (ESA) in particu-
lar. On the other hand, the nature of the problems has changed. In product development,
there was a time when ESA provided the solution directly, for example, the stress at a
point or the failure load. In research, there was a time when ESA gave insight into the
phenomenon, for example, dynamic crack initiation and arrest. What they both had in
common is that they attempted to give “the answer”; in short, we identified an unknown
and designed an experiment to measure it. Modern problems are far more complex, and
the solutions required are not amenable to simple or discrete answers.
Intruth,experimentalengineershavealwaysbeeninvolvedinmodelbuilding,butthe
natureofthemodelhaschanged.Itwasoncesufficienttomakeatable,listingdimensions
andmaterialproperties,andsoon,ormakeagraphoftherelationshipbetweenquantities,
and these were the models. In some cases, a scaled physical construction was the model.
Nowadays the model is the FEM model, because, like its physical counterpart, it is a
dynamicmodelinthesensethatifstressesorstrainsordisplacementsarerequired,these
arecomputedontheflyfordifferentloads;itisnotjustadatabaseofnumbersorgraphs.
Actually, it is even more than this; it is a disciplined way of organizing our current
knowledge about the structure or component. Once the model is in order or complete, it
can be used to provide any desired information like no enormous data bank could ever
do; it can be used, in Hamilton’s words, “to utter its revelations of the future”. It is this
predictive and prognostic capability that the current generation of models afford us and
that traditional experimental stress analysis is incapable of giving.
x Preface
Many groups can be engaged in constructing the model; there is always a need for
new elements, new algorithms, new constitutive relations, or indeed even new computer
hardware/software such as virtual reality caves for accessing and displaying complex
models. Experimental stress analysts also have a vital role to play in this.
That the model is the focal point of modern ESA has a number of significant impli-
cations. First, collecting data can never be an end in itself. While there are obviously
some problems that can be “solved” using experimental methods alone, this is not the
norm. Invariably, the data will be used to infer indirectly (or inversely as we will call it)
something unknown about the system. Typically, they are situations in which only some
aspects of the system are known (geometry, material properties, for example), while
otheraspectsareunknown(loads,boundaryconditions, behaviorofa nonlinearjoint, for
example) and we attempt to use measurements to determine the unknowns. These are
whatwecallpartiallyspecifiedproblems.Thedifficultywithpartiallyspecifiedproblems
is that, far from having no solution, they have great many solutions. The question for
us revolves around what supplementary information to use and how to incorporate it in
thesolutionprocedure.Whichbringsustothesecondimplication.Theengineeringpoint
to be made is that every experiment or every experimental stress analysis is ultimately
incomplete;therewillalwaysbesomeunknowns,andatsomestage,thequestionofcop-
ing with missing information must be addressed. Some experimental purists may argue
that the proper thing to do is to go and collect more data, “redo the experiment,” or
designabetterexperiment.Butwereiteratethe pointthateveryexperiment(whichdeals
with a real structure) is ultimately incomplete, and we must develop methods of coping
with the missing information. This is not a statistical issue, where, if the experiment
is repeated enough times, the uncertainty is removed or at least characterized. We are
talking about experimental problems that inherently are missing enough information for
a direct solution.
A final point: a very exciting development coming from current technologies is the
possibility of using very many sensors for monitoring, evaluation, and control of engi-
neering systems. Where once systems were limited to a handful of sensors, now we can
envisage using thousands (if not even more) of sensors. The shear number of sensors
opens up possibilities of doing new things undreamt of before, and doing things in new
ways undreamt of before. Using these sensors intelligently to extract most information
falls into the category of the inverse problems we are attempting to address. That we
can use these sensors in combination with FEM-based models and procedures for real-
time analysesof operatingstructuresor post analyses of failedstructures is anincredibly
exciting possibility.
With some luck, it is hoped that the range of topics covered here will help in the
realization of this new potential in experimental mechanics in general and experimental
stress analysis in particular.
Lafayette,Indiana JamesF. Doyle
December,2003