SUMMARY Spline and generalized spline smoothing is shown to be equivalent to Bayesian estimation with a partially improper prior. This result supports the idea that spline smoothing is a natural solution to the regression problem when one is given a set of regression functions but one also wants to hedge against the possibility that the true model is not exactly in the span of the given regression functions. A natural measure of the deviation of the true model from the span of the regression functions comes out of the spline theory in a natural way. An appropriate value of this measure can be estimated from the data and used to constrain the estimated model to have the estimated deviation. Some convergence results and computational tricks are also discussed.

Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression

The behavior of solids can be represented by a variety of continuum theories. For example, Cosserat elasticity allows the points in the continuum to rotate as well as translate, and the continuum supports couple per unit area as well as force per unit area. We examine experimental methods for determining the six Cosserat elastic constants of an isotropic elastic solid, or the six Cosserat relaxation functions of a Cosserat viscoelastic solid. We also consider other generalized continuum theories (including micromorphic elasticity, Cowin's void theory, and nonlocal elasticity). Ways of experimentall y discriminating among various generalized continuum representations are presented. The applicability of Cosserat elasticity to cellular solids and fibrous composite materials is considered as is the application of related generalized continuum theories. I Introduction The classical theory of elasticity is presently used in engineering analyses of deformable objects at small strain. However there are other continuum theories for linear isotropic materials. Some have more freedom, and some have less freedom than classical elasticity. The various continuum theories are all mathematically self consistent. Therefore a discrimination among them is to be made by experiment. It is the purpose of this article to explore the physical consequences of various continuum theories, and how these consequences may be used in the design of experiments to discriminate among the theories. The constitutive equations for several theories are presented, and some of the salient consequences of each theory are stated and discussed. Some of the causal physical mechanisms associated with each theory are briefly discussed. Experimental methods for evaluating materials as generalized continua are presented, with emphasis on Cosserat elasticity. The treatment is restricted to linearly elastic behavior; study of Cosserat plasticity and related issues is presented elsewhere in this volume. A discussion of experimental aspects of generalized continua is considered particularly appropriate in view of the fact that most of the work done thus far in generalized continuum mechanics has been theoretical.

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Experimental methods for study of cosserat elastic solids and other generalized elastic continua

This paper describes a new image-shearing camera which focuses two laterally sheared images at the film plane. With coherent illumination, this camera becomes a shearing interferometer, which directly measures the derivatives of the surface displacements. This strain measuring tool enjoys several advantages over the conventional, holographic, and speckle interferometry, namely, (1) better fringe quality (than speckle interferometry); (2) does not require special vibration isolation; (3) very simple optical setup; (4) direct determination of strains; and (5) extended controllable range of sensitivity.

Image-shearing camera for direct measurement of surface strains.

This article describes the experimental investigation of Cosserat (or micropolar) elasticity and surface damage effects in closed-cell polymethacrylimide foams of different densities. The method of size effects was used to find the degree of Cosserat behaviour for both cylindrical and square cross-section specimens in bending and torsion. The foams were found to behave as Cosserat materials in which slender specimens appear less stiff than thick ones, provided sufficient care is taken when machining the specimens. Surface damage caused by the machining process may cause the apparent stiffness to decrease with decreasing specimen size, giving an opposite softening size effect.

/pdf/size-effects-due-to-cosserat-elasticity-and-surface-damage-4f7t2tctp3.pdf

Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam

An approximate solution in the form of a polynomial is presented for the normal stress distribution adjacent to a circular hole in an infinite orthotropic plate. Comparison of the approximate solut...

Approximate Stresses in an Orthotropic Plate Containing a Circular Hole

An improved speckle-shearing interferometric method is presented that allows simultaneous determination of derivatives of surface displacements of a structure with respect to four different directions. The technique relaxes several limitations associated with conventional interferometry and thus is adaptable to nonlaboratory environments. The relevant theory is presented, and the method demonstrated by determining spatial derivatives of in-plane and out-of-plane displacements of statically loaded and vibrated structures.

Speckle-Shearing Interferometric Technique: a Full-Field Strain Gauge

Experimental and finite-element analyses are presented for the anisotropic states of stress, strain and fracture of a glass-epoxy plate containing a circular hole and subjected to uniaxial tension. Strains were experimentally measured using foil gages, moire and birefringent coating. Stresses are computed in the linear range from the measured strains. While the hole reduces the plate strength by a factor of two, the maximum tensile strain at fracture is greater than the ultimate strain in a plate without a hole. Fracture consists of crack initiation at the hole boundary but off the horizontal axis. Away from the hole, failure is accompanied by considerable delamination. Discontinuous crack propagation is present.

Stress and failure analysis of a glass-epoxy composite plate with a circular hole

Moire techniques were developed, adapted and applied to the determination of strain fields in filamentary composite laminates. Conventional techniques, using 1000 line-per-inch (Ipi) arrays bonded or photoprinted onto the specimen, were applied to glass-epoxy and boron-epoxy specimens with holes and cracks. Techniques for tenfold fringe multiplication were also applied to glass-epoxy and boron-epoxy laminates with holes. A reflective surface of glass-like smoothness was produced on the specimen and a 500 dot-per-inch grid photoprinted on it. A rigid distortion-free camera was used for recording replicas of the specimen grid by projection photography. These replicas were analyzed with a 200 line-per-millimeter (5080 Ipi) grating for reconstruction of moire-fringe patterns. These patterns were analyzed by graphical and mechanical differentiation using second-order moire. Strain distributions and strain-concentration factors were in very good agreement with theoretical and other experimental results.

Strain analysis of composites by moiré methods - Procedures and results using conventional and fringe-multiplication techniques are described and illustrated

Cubic-spline and discrete-quadratic polynomial techniques are presented for reliably computing up to third-order derivatives of experimental information. The concept is demonstrated by stress analyzing from measured displacements a transversely loaded plate and a beam under four-point bending. The respective displacement fields were recorded using holography and moire. The accuracy of the employed numerical-differentiation techniques is indicated.

https://link.springer.com/content/pdf/10.1007/BF02323967.pdf

Higher-order numerical differentiation of experimental information - Cubic-spline and discrete-quadratic polynomials are described for numerically computing up through third-order derivatives. Concept is demonstrated by stress analyzing, from moiré and holographically recorded displacements, loaded plates and beams

Photomechanical techniques are developed for studying wave propagation in anisotropic media such as fiber-reinforced composites and rocks. Moire fringes in transparent and opaque materials, and isochromatic fringes in transparent composites, are photographed at rates up to over one-million frames/second. A Beckman and Whitley 189 framing camera is utilized with illumination provided by argon flashbombs or a special electronic flash.

R. E. Rowlands

Papers

Speckle-Shearing Interferometric Technique: a Full-Field Strain Gauge

Stress and failure analysis of a glass-epoxy composite plate with a circular hole

Strain analysis of composites by moiré methods - Procedures and results using conventional and fringe-multiplication techniques are described and illustrated

Wave motion in anisotropic media by dynamic photomechanics.