Tikhonov regularization with Incremental Hole-Drilling and the Integral Method in Cross-Ply Composite Laminates

Background: Incremental hole-drilling with the integral method has been extensively used in composite laminates but is sensitive to small measurement errors. Error sensitivity can be reduced by limiting the number of depth increments used in the calculation procedure. This approach is limited if a rapidly varying residual stress distribution exists since the calculated stress in each incremental depth is considered constant. Distortion of stress results can consequently occur due to averaging effects if the depth increments become too large. Tikhonov regularization is usually applied in isotropic materials to smooth the resulting residual stress distribution and reduce stress uncertainties, but has only been applied to composite laminates using the slitting technique. Objective: The intention of this work is to extend the use of Tikhonov regularization to incremental hole-drilling of composite laminates using the integral method. Methods: Finite element modelling is used to calculate the necessary calibration coefficients for unit pulses of uniform stress. Monte Carlo simulation is used to the determine uncertainties in the calculated residual stress distributions. Tikhonov regularization is optimised to reduce the stress uncertainty, while ensuring that the stress solution is not distorted. Results: The method is demonstrated on a GFRP (Glass fibre reinforced plastic) laminate of [0₂/90₂]_s construction and the calculated residual stress field is compared with those obtained by the standard integral method and series expansion. Conclusions: It is found that Tikhonov regularization significantly improves the accuracy of the standard integral method in composite laminates and shows good agreement with the series expansion method.

     

Effect of laminate construction on residual stresses in graphite/polyimide composites

The influence of ply-stacking sequence and ply orientation on the magnitude of lamination residual stresses in graphite/polyimide angle-ply laminates was investigated. The effect of stacking sequence was investigated with laminates of [0₂/±45]_s, [±45/0₂]_s, [0/+45/0/?45]_s and [+45/0₂/?45]_s layup. The effect of ply orientation was evaluated with additional specimens of [0₂/±15]_s and [0₂/90₂]_s layup. Thermal strains were measured using embedded-gage techniques. Residual strains were determined by comparing thermal strains in the angle-ply laminates with those of a unidirectional laminate. The ply-stacking sequence did not have an influence on the magnitude of residual strains. The highest residual strains occur in the [0₂/90₂]_s laminate and the lowest, approximately one-fourth in magnitude, occur in the [0₂/±15]_s laminate. The maximum residual strains in the [0₂/±45]_s group are slightly lower than those in the [0₂/90₂]_s laminate. Residual stress computations show that, at room temperature, the transverse-to-the-fibers stresses in all plies, except those of the [0₂/±15]_s laminate, exceed the transverse tensile strength of the unidirectional material.     

Validation of Mechanical Strain Relaxation Methods for Stress Measurement

Most validation studies of mechanical strain relaxation (MSR) methods for residual stress measurement rely on using the saw-tooth residual stress distribution resulting from four point bending and elastic–plastic deformation. Validation studies using simple applied stress profiles in rectangular steel beams are used in this work, together with beams subjected to elastic–plastic bending. Two MSR methods are explored, deep-hole drilling (DHD) and incremental centre hole drilling (ICHD). As well as a series of experiments, finite element analyses are conducted to determine the accuracy in the inversion of measured deformation to reconstruct stress. The validation tests demonstrated that apart from the applied stresses, the initial residual stresses also contribute even when samples are expected to be stress free. The uncertainty in measurement for the two MSR methods is determined, with the uncertainty in near surface measurement found to be significantly larger than uncertainty for interior measurement. In simple loading cases (and simple stress profiles) the uncertainty in measurement and hence the degree of validation is shown to be within about ±50 MPa for steel for “known” stress up to about 140 MPa. However, if the residual stress distribution is more complex there arises increased uncertainty in the predicted residual stress and lack of confidence between measurements methods.     

Singular full-field stresses in composite laminates with open holes

A method of the superposition of a hybrid and displacement approximation was developed to provide the accurate stress fields in a multilayered composite laminate, including the singular neighborhood of the ply interface and the hole edge. Asymptotic analysis was used to derive the hybrid stress functions. The displacement approximation is based on the polynomial B-spline functions. The method provides the determination of the coefficient of the singular term along with convergent stress components including the singular regions. Reissner’s variational principle was employed. Simple [45/−45]_s and quasi-isotropic IM7/5250 [45/90/−45/0]_s laminates were analyzed. Uniaxial loading and residual stress calculation (quasi-isotropic laminate) were considered. A convergence study showed that accurate values of the coefficient of the singular term of the asymptotic stress expansion could be obtained with coarse out-of-plane and in-plane subdivisions. The interaction between the singular terms on the neighboring interfaces was found to be important for the convergence with coarse subdivisions. Converged transverse interlaminar stress components as a function of the distance from the hole edge, were shown for all examples.     

Micron-Scale Residual Stress Measurement by Micro-Hole Drilling and Digital Image Correlation

This paper reports a new technique, namely the incremental micro-hole-drilling method (IμHD) for mapping in-plane residual or applied stresses incrementally as a function of depth at the micron-scale laterally and the sub-micron scale depth-wise. Analogous to its macroscale counterpart, it is applicable either to crystalline or amorphous materials, but at the sub-micron scale. Our method involves micro-hole milling using the focused ion beam (FIB) of a dual beam FEGSEM/FIB microscope. The resulting surface displacements are recorded by digital image correlation of SEM images recorded during milling. The displacement fields recorded around the hole are used to reconstruct the stress profile as a function of depth. In this way residual stresses have been characterized around a drilled hole of 1.8microns. diameter, enabling the profiling of the stress variation at the sub-micron scale to a depth of 1.8 microns. The new method is used to determine the near surface stresses in a (peened) surface-severe-plastically-deformed (S²PD) Zr₅₀Cu₄₀Al₁₀ (in atomic percent, at.%) bulk metallic glass bar. In plane principal stresses of -800 MPa ± 90 MPa and −600 MPa ± 90 MPa were measured, the maximum compressive stress being oriented 15° to the axis of the bar.     

Thermal residual stresses in thick graphite/epoxy composite laminates—Uniaxial approach

Thermal residual stresses have been known to be very large in laminates of continuous-fiber-reinforced polymer composites. When the thickness of the laminate is large, however, the measurement of the residual stresses raises questions on the accuracy of the conventional methods. A novel concept of layer separation is developed to measure quantitatively and precisely the tensile residual stresses in thick plates with layered distribution of residual stresses. It is applied to thick [O₂/9O₄]_13s, AS/3501-6 graphite/epoxy laminates. The test specimens were mechanically modeled into the thin strips for the application of the new concept of layer separation. The tensile residual stresses measured in the 90-deg layers of these laminates are nonuniform throughout the specimen, and vary from 55.6 MPa to 71.4 MPa. It is very interesting to compare these values with the transverse strengthF ₂ ^tu of AS/3501-6 unidirectional composites, which is 65.4 MPa.     

A PREDICTIVE APPROACH TO THE IN-PLANE MECHANICAL PROPERTIES OF STITCHED COMPOSITE LAMINATES

This contribution attempts to model the alteration of the in-plane elastic properties in laminates caused by stitching,and to predict the in-plane effective tensile strength of the stitched composite laminates.The distortion of in-plane fibers is considered to be the main cause that affects the in-plane mechanical properties.A fiber distortion model is proposed to characterize the fiber misalignment and the fiber content concentration due to stitching.The undistorted region,the fiber distortion region,the resin-rich pocket and the through-thickness reinforcement section are taken into account.The fiber misalignment and inhomogeneous fiber content due to stitching have been formulated by introducing two parameters,the distortion width and maximum misalignment.It has been found that the ply stress concentration in stitched laminates is influenced by the two concurrent factors,the stitch hole and inhomogeneous fiber content.The stitch hole brings about the stress concentration whereas the higher fiber content at the local region induced by stitching restrains the local deformation of the composite.The model is used to predict the tensile strength of the [0/45/0/-45/90/45/0/-45]_(2s) T300/QY9512 composite laminate stitched by Kevlar 29 yarn with different stitching configurations,showing an acceptable agreement with experimental data.     

材料非线性对复合材料层合板热自由边界效应的影响

本文用准三维有限元法研究了材料非线性对复合材料层合板热自山边界效应的影响,给出了修正型Hahn-Tsai非线性应力-应变关系的三维形式。由本文非线性分析方法得到的层间应力与以往由线性分析方法得到的层间应力做了比较,结果表明:材料非线性能显著降低层间剪应力的集中程度,但对层间正应力影响不太明显。     

Residual Stress Analysis on Thick Film Systems by the Incremental Hole-Drilling Method – Simulation and Experimental Results

The residual stress state in thick film systems, as for example thermal spray coatings, is crucial for many of the component’s properties and for the evaluation of the integrity of the coating under thermal and/or mechanical loading. Therefore it is necessary to be able to determine the local residual stress distribution in the coating, at the interface and in the substrate. The incremental hole-drilling method is a widely used method for measuring residual stress depth profiles, which was already applied for thermally sprayed coatings. But so far no reliable hole-drilling evaluation method exists for layered materials having a stress gradient in depth. The objective was to investigate, how far existing evaluation methods of the incremental hole-drilling method that are only valid for residual stress analysis of homogenous material states can be applied to thick film systems with coating thicknesses between 50 μm and 1000 μm and to point out the application limits for these already existing methods. A systematic Finite Element (FE) study was carried out for coating systems with an axisymmetric residual stress state σ₁?=?σ₂. It is shown that conventional evaluation methods developed for homogeneous, non-layered material states can be successfully applied for a stress evaluation in the substrate and the coating for small and for sufficiently large coating thicknesses, respectively, regardless of the type of evaluation algorithm used i.e. the differential or the integral method. The same accounts for material combinations that have a Young’s modulus ratio close to one, between 0.8 and 1.2. The studies indicated that outside the given ranges case specific calibration must be applied to calculate reliable results. Further, calibration data were determined case specifically for a selected model coating system. The accuracy of a residual stress determination using these case specific calibration data was examined and the sensitivity of the evaluation with respect to an accurate knowledge of the boundary conditions of the coating system i.e. the coating thickness and the Young modulus was studied systematically. Finally, the calibration data were applied on a thermally sprayed aluminium coating on a steel substrate analysis and the results for using the incremental hole drilling method were compared to results from X-ray stress analysis.     

复合材料层合板粘弹性固化残余应力分析

基于Ｓｃｈａｐｅｒｙ积分型粘弹性本构关系，推导了考虑横向剪切效应的复合材料层合板线性热粘弹性有限元分析列式，对层合板的粘弹性响应和加工成型过程中的残余应力进行了分析，给出一些有意义的结果     

Measurement and Prediction of Residual Stresses in Quenched Stainless Steel Components

Austenitic stainless steel cylinders and rings are spray water quenched to create residual stresses at or greater than the yield strength. The residual stresses are measured using neutron diffraction, and two mechanical strain relaxation methods: deep hole drilling and incremental centre hole drilling. This paper compares the measurements with predictions of quenching using finite element analysis. Also finite element analysis is used to mimic deep hole and incremental centre hole drilling methods and to reconstruct residual stresses as if they have been measured. The measurements reveal similar trends to the predictions but there is only limited agreement between their magnitudes. However, there is better agreement between the reconstructed stresses and the measurements. Both the two mechanical strain relaxation methods reveal that large discrepancies occur between measurements and predictions arise because of plasticity. Irrespective of this and surprisingly there is good agreement between deep hole drilling and neutron diffraction measurements.     

Residual Stress Analysis of Orthotropic Materials by the Through-hole Drilling Method

The present study deals with the development and the application of the through-hole drilling method for the residual stress analysis in orthotropic materials. Through a systematic theoretical study of the stress field present on orthotropic plates with a circular hole, the relationships between the relaxed strains measured by a rectangular strain gauge rosette and the Cartesian components of the unknown residual stresses are obtained. The theoretical formulas of each influence coefficient allow the user an easy application of the method to the analysis of uniform-residual stresses on a generic homogeneous orthotropic material. Furthermore, to extend the method to the analysis of the residual stresses on orthotropic laminates, caused by initial in-plane loadings, an alternative formulation is implemented. The accuracy of the proposed method has been assessed through 3D numerical simulations and experimental tests carried out on unidirectional, cross-ply and angle-ply laminates.     

Strain and stress concentrations in composite laminates containing a hole

The strain concentrations of orthotropic composite laminates containing a circular hole and subject to tensile loading were measured experimentally using strain gages. Then the stress concentrations were calculated using the strain distributions in the initial region of the stress-strain curve before any microdamages were developed. The graphite/epoxy AS4/3502 [O₂/±45]_2s and [45]_4s were chosen to represent fiber-dominated and matrix-dominated laminates, respectively. Several combinations of hole-diameter/plate-width ratio were designed to show the width effect. The conditions of the laminates, after the holes were drilled, were examined using X-ray techniques. Good correlation was obtained between theory and experimental result using specimens in good condition (without machining damages). A procedure for accurately determining the strain and stress concentrations is given. Paper was presented at the 1989 SEM Spring Conference on Experimental Mechanics, held in Cambridge, MA on May 29–June 1.     

Stress intensity factors and crack initiation directions for ceramic/metal joint

Four points bending tests for Si₃N₄/Cu/S45C joint specimen showed that the bending strengths depend on the residual stresses that originated from joining process. The residual thermal stresses caused an edge sub-interface crack to initiate in the ceramic. The stress intensity factors (SIFs) of the edge sub-interface crack located at distance h from the interface with or without interlayer metal were calculated by the Green's function obtained from a finite element analysis. The crack path at the joint specimen under four points bending loading with the influence of residual stresses was also evaluated by the maximum tensile stress criterion. Finally the effect of residual stress on the crack path was found numerically; the interlayer metal decreases the deflection angle of crack from interface by reducing the residual stress.     

Fiber fracture and strength degradation in unidirectional graphite/epoxy composite materials

An experimental study was conducted to determine the influence of load factor on fiber fracture development and residual strength of fatigue loaded unidirectional graphite/epoxy composite laminated. 8-phy composite laminates with a layer of release cloth imbedded at the middle ply were fatigued at different load levels and were examined for fiber fracture and residual strength at several stages of life based on the average number of cycles to failure (according to S-N data). From the experimental results, it is evident that the number of fiber fractures is nearly constant after the first few percent of the life. It is also suggested that the load level is much more important than the number of cycles of loading in the determination of the state of fiber fracture. This behavior was interrupted at high load levels (S60% S_u where the final fracture was highly affected by the longitudinal matrix splittings. Residual strength is found to be independent of the global fiber fracture density, and to be controlled by local behavior such as matrix cracking, local clustering of fiber fractures, and other local stress concentrations.     

Measuring Inaccessible Residual Stresses Using Multiple Methods and Superposition

The traditional contour method maps a single component of residual stress by cutting a body carefully in two and measuring the contour of the cut surface. The cut also exposes previously inaccessible regions of the body to residual stress measurement using a variety of other techniques, but the stresses have been changed by the relaxation after cutting. In this paper, it is shown that superposition of stresses measured post-cutting with results from the contour method analysis can determine the original (pre-cut) residual stresses. The general superposition theory using Bueckner’s principle is developed and limitations are discussed. The procedure is experimentally demonstrated by determining the triaxial residual stress state on a cross section plane. The 2024-T351 aluminum alloy test specimen was a disk plastically indented to produce multiaxial residual stresses. After cutting the disk in half, the stresses on the cut surface of one half were determined with X-ray diffraction and with hole drilling on the other half. To determine the original residual stresses, the measured surface stresses were superimposed with the change stress calculated by the contour method. Within uncertainty, the results agreed with neutron diffraction measurements taken on an uncut disk.     

Incremental Hole Drilling for Residual Stress Analysis of Thin Walled Components with Regard to Plasticity Effects

The incremental hole-drilling method is widely used in residual stress depth distribution analysis. However, two specific difficulties with the generalization of the incremental method exist, including the consideration of the sample thickness and residual stress states close to the local material’s yield strength. The stress concentration effect of the hole can lead to plastic deformation in the vicinity of the hole, which results in an overestimation of residual stresses. Typically, the effect of the component’s thickness and the plasticity effects are analyzed separately and correction approaches are proposed. In the current paper, we analyze the combined effects of plasticity and thickness on residual stress analysis using the incremental hole-drilling method. A systematic study was performed on steel samples with (i) isotropic and (ii) anisotropic elastic and elasto-plastic material behavior with varying thicknesses ranging between 1 mm and 4 mm. Electronic speckle pattern interferometry (ESPI) hole-drilling experiments were conducted on beam samples loaded using a 4-point bending fixture. Finite element simulations were conducted to gain insight into the effects of incremental hole-drilling. The results indicate that reducing the component’s thickness increases the plastic deformation in the vicinity of the hole and results in significant stress deviations. Thin components bend during hole-drilling as a result of the loss of stiffness, which amplifies the plasticity effect.     

An experimental investigation on the tensile moduli and strengths of graphite/epoxy laminates

The results of a series of tensile tests on graphite/epoxy [0°]_8S and [0°/±30°/0°]_2S laminates at rates varying from 0.002 in/min to 2 in./min are reported. The loads are applied at various angles to the fiber directions in each case. The rate-dependent behavior of the stress-strain response is assessed. Evidence is presented to indicate that failure first occurs on inner plies. Also, evidence is presented to indicate that, in some cases, moduli increase with increased stress or strain level. Lamination theory is used to predict moduli and comparisons with experiment are given. This theory is also used in conjunction with three failure theories to predict ultimate strengths with varying degrees of success. Further, two approaches to ply unloading after first-ply failure are used and discussed. One is a standard method found in the literature while the other is a proposed ‘strength-of-materials’ type of technique which is computationally much simpler.     

On the measurement of residual-stress gradients in aluminum-alloy specimens

The assessment or prediction of fatigue life or strength improvement due to residual stresses requires knowledge of their magnitude and distribution. This paper presents an extension of the modified hole-drilling technique (MHDT) to the measurement of stress gradients in a biaxial-residual-stress field. This is achieved by taking a series of ‘point’ measurements and evaluating the stress profile with due consideration to the effects of hole location, the interaction between holes and the redistribution of stress due to hole drilling. An application to the measurement of residual stresses induced in 2024-T3 aluminum-alloy specimens by edge-dimpling technique is described and the method of compensation for the effect of redistribution of stress is explained. The experimental results are shown to be in good agreement with those obtained elsewhere by an analytic-numerical solution.     

Bayesian Identification of Elastic Constants in Multi-Directional Laminate from Moiré Interferometry Displacement Fields

The ply elastic constants needed for classical lamination theory analysis of multi-directional laminates may differ from those obtained from unidirectional laminates because of three dimensional effects. In addition, the unidirectional laminates may not be available for testing. In such cases, full-field displacement measurements offer the potential of identifying several material properties simultaneously. For that, it is desirable to create complex displacement fields that are strongly influenced by all the elastic constants. In this work, we explore the potential of using a laminated plate with an open-hole under traction loading to achieve that and identify all four ply elastic constants (E ₁ , E ₂ , ν ₁₂ , G ₁₂ ) at once. However, the accuracy of the identified properties may not be as good as properties measured from individual tests due to the complexity of the experiment, the relative insensitivity of the measured quantities to some of the properties and the various possible sources of uncertainty. It is thus important to quantify the uncertainty (or confidence) with which these properties are identified. Here, Bayesian identification is used for this purpose, because it can readily model all the uncertainties in the analysis and measurements, and because it provides the full coupled probability distribution of the identified material properties. In addition, it offers the potential to combine properties identified based on substantially different experiments. The full-field measurement is obtained by moiré interferometry. For computational efficiency the Bayesian approach was applied to a proper orthogonal decomposition (POD) of the displacement fields. The analysis showed that the four orthotropic elastic constants are determined with quite different confidence levels as well as with significant correlation. Comparison with manufacturing specifications showed substantial difference in one constant, and this conclusion agreed with earlier measurement of that constant by a traditional four-point bending test. It is possible that the POD approach did not take full advantage of the copious data provided by the full field measurements, and for that reason that data is provided for others to use (as on line material attached to the article).