Compliance Equations Research Articles

Numerical investigation of compliance equations used in the R‐curve testing for clamped SE(T) specimens

AbstractThree‐dimensional finite element analysis of clamped single‐edge tension specimen is performed to examine the accuracy of the crack‐mouth opening displacement compliance equations reported in the literature. The analysis covered both plane‐sided and side‐grooved specimens with a wide range of specimen configurations including nine relative crack lengths (a/W) ranging from 0.1 to 0.9 and seven relative thicknesses (B/W) ranging from 0.25 to 4. Based on the finite element analysis results, the crack length/compliance modification factors (ρ(a/W) and ρ(BCE)) are developed to improve the accuracy of the compliance equations. The results of this study can improve the accuracy of the experimentally determined J‐R and CTOD–R curves using the single‐specimen technique and single‐edge tension specimens.

A new approach for determining the R-curve in DCB tests without optical measurements

A new method for obtaining the R-curve point to point by the Double Cantilever Beam test is proposed. Besides the determination of new compliance and energy release rate equations, the analytical model presented leads to calculate the crack length for every pair of load and displacement values, without any optical measurement. In addition, a simple trigonometric approximation is proposed for the calculation of large displacement effects. The approach is checked with experimental results.

Finite element analysis of crack mouth opening displacement compliance in crack length evaluation for clamped single edge tension specimens

ABSTRACTIn the unloading compliance method developed for clamped single edge tension (SE(T)) specimens, six crack mouth opening displacement (CMOD)‐based compliance equations (i.e. a/W = f(BCE′)) were proposed for the crack length evaluation without clearly clarifying the corresponding predictive accuracies. In addition, the effective elastic modulus (Ee) that reflects the actual state of stress should also be introduced in the crack length evaluation for SE(T) specimens, because the actual state of stress in the remaining ligament of the test specimen is neither plane stress (E) nor plane strain (E′). In this study, two‐dimensional (2D) plane strain and three‐dimensional (3D) finite element analyses (FEAs) are carried out to investigate predictive accuracies of the six compliance equations. In both 2D and 3D FEA, specimens with a wide range of crack lengths and geometric configurations are included. For a given specimen, the value of Ee that presents the equivalent stress state in the remaining ligament is calculated on the basis of 3D FEA data. A set of formulae for the clamped SE(T) specimen is proposed that allows to evaluate Ee from the corresponding CMOD compliance. This approach is verified using numerical data. The observations of the numerical verification suggest that the use of Ee instead of E or E′ in CMOD‐based compliance equations markedly improves the accuracy of the predicted crack length for clamped SE(T) specimens.

A Numerical Study on Effective Modulus of Elasticity in Crack Length Evaluation for Single-Edge Bending Specimens

Abstract The actual state of stress in the remaining ligament of a single-edge bending (SE(B)) specimen, which is neither plane stress nor plane strain, is critical in accurately predicting the crack length in the specimen involved in the unloading compliance method. Three-dimensional finite element analyses were carried out in this study to evaluate the effective modulus of elasticity that reflects the actual state of stress corresponding to the crack mouth opening displacement (CMOD) compliance of the SE(B) specimen. Both plane-sided and side-grooved specimens with a wide range of geometric configurations and crack lengths were included in the study. A practical approach is further proposed to quickly evaluate the effective modulus from the CMOD compliance determined based on the test data. The proposed approach was verified using experimental data and numerical analysis results. The results of the verification suggest that replacing the elastic modulus corresponding to the plane stress or plane strain condition with the effective modulus in commonly used a/W–CMOD compliance equations can markedly improve the accuracy of the predicted crack length.

A Study on the Fitting Accuracy of CMOD Compliance Equations

Abstract The maximum error of the inverse fitting of Tada et al.’s equation of the a/W–crack mouth opening displacement (CMOD) compliance equation proposed by Joyce and adopted in ASTM E1820-11 for single edge bend [SE(B)] specimens with shallow cracks (0.05 ≤ a/W < 0.45) is observed to be 2.03 % for 0.05 < a/W < 0.1. A new a/W–CMOD compliance equation with the same fifth-order polynomial fitting applicable for SE(B) specimens with 0.05 ≤ a/W ≤ 0.7 is therefore proposed. The maximum fitting error of the proposed equation is approximately 0.19 %.

Closed-form compliance equations for power-function-shaped flexure hinge based on unit-load method

A new type of flexure hinge, called power-function-shaped flexure hinge, is presented in this paper. Based on the unit-load method, the closed-form compliance equations of the flexure hinge are derived, and the motion accuracy is investigated. These equations are verified through finite element analysis, which shows that the maximum error is less than 15%. Based on the compliance equations, numerical simulations are conducted to discuss the performance of the flexure hinge. The results indicate that, the stiffness of the power-function-shaped flexure hinge decreases as r increases, increases as β increases, and increases as t increases; the displacement along axis y is the dominant parasitic motion; and based on the same l, h and b, the power-function-shaped flexure hinge can rotate more accurately than circular flexure hinge with the same β and V-shaped flexure hinge with the same β and r. The equations obtained can be utilized to design the power-function-shaped flexure hinges, and can achieve higher motion accuracy than the commonly used circular flexure hinge and V-shaped flexure hinge.

Study on Delamination Propagation Behavior and Measurement Method for One Kind of New Fiber Metal Laminates

The delamination at the interface between the aluminium and prepreg layers has greater influence on the fatigue crack growth in fiber metal laminates. To study delamination propagation property of one kind of new fiber metal laminates, by using 2/1 lay-up specimen, this article accomplished delamination propagation test under various stress levels at R=0.1 by optical method and compliance method respectively. The delamination propagation property of this laminate is obtained by analyzing and calculating. A modified compliance equation is created to calculate the delamination size of this laminate more accurately by adding a non-dimensional factor.

Comparative Analysis of Parabolic and Hyperbolic Flexure Hinges Using Closed-Form Equations

The comparative analysis of a parabolic and hyperbolic flexible hinges were presented. The closed-form compliance equations in the working and non-working plane can be derived by using the Castigliano’s second theorem. In order to compare the relative performance of parabolic and hyperbolic flexure hinges, a ratio function between specific compliances of the two hinges is defined. We analyzed the non-dimensional ratios of working plane and non-working plane compliance to show the relative properties of parabolic versus hyperbolic flexure hinges. The analysis results has showed compared to a hyperbolic flexure, a parabolic flexure can produce more desired output and is slightly less sensitive to axial effects under identical loading conditions. Meanwhile, the comparative study on the parabolic and hyperbolic flexure hinges can provide a theoretical basis for choosing them.

Performance of a Parabolic Flexible Joint Using Compliance Closed-Form Equation

Considering the effects of structural sizes of a parabolic flexible joint on its performance, compliance, sensitivity and accuracy analysis are performed in this paper. The closed-form equations are formulated for compliances to characterize both the active rotation and all other in- and out-of-plane parasitic motions, and accuracy equations are also obtained by using the Castigliano’s second theorem. According to the closed-form compliance equations, the effects of the structural parameters, such as thickness, notch depth, length and width, on the compliance of flexible joint are analyzed, respectively. In order to explore the effects of variations of the structural parameters on its compliance, the sensitivity analysis is carried out. And the accuracy analysis of the parabolic flexure joint is performed. The analysis results have shown that the parameters have large influence on in-plane compliance, and have small effects on out-of-plane one. And changes of compliance are larger within initial small range of parameters, while it has small variation in larger size range. And the length of joint has no effect of sensitivity

Interface Fracture Assessment on Sandwich DCB Specimens

This article examines the fracture behavior of sandwich type double cantilever beam (DCB) specimens under monotonic and cyclic loading conditions. The specimen consists of two layers of OFHC copper and one layer of ViaLux 81 photo-definable dry film. The ViaLux 81 is placed in between copper layers. Analytical expression for the strain energy release rate (GI) of the sandwich DCB specimen is derived by following the plate theory based model and compliance method. Finite element analysis has also been carried out on the specimen and the load displacement results for the specified crack or delamination length are obtained. The compliance equation in terms of the delamination length is derived for the specimens for obtaining the strain energy release rate (GI). From the generated R-curve of the interface, the failure load is estimated for the specified delamination length. From the crack growth data, the number of cycles to failure under cyclic loading is estimated for the initial delamination length.

Closed-form compliance equations of filleted V-shaped flexure hinges for compliant mechanism design

This paper presents the closed-form compliance equations for the filleted V-shaped flexure hinges. The in-plane and out-of-plane compliances of the flexure hinges are developed based on the Castigliano's second theorem. The accuracy of motion, denoted by the midpoint compliance of the flexure hinges, is also derived for optimized geometric design. The influences of the geometric parameters on the characteristics of the flexure hinges are investigated. It is noted that the filleted V-shaped flexure hinges have diverse ranges of compliance corresponding to different filleted radius R and angle θ. These types of hinges can provide both higher and lower stiffnesses than circular flexure hinges. This makes filleted V-shaped flexure hinges very useful for wide potential applications with different requirements. The finite element analysis is used to verify the established closed-form compliance equations for these filleted V-shaped flexure hinges.

A new method for determining mode II R-curve by the End-Notched Flexure test

A new method for obtaining the mode II R-curve in a End-Notched Flexure test is proposed in the present work. New compliance and energy release rate equations have been derived incorporating shear, local deformation and bending rotation effects. Mode II R-curve, which represents energy release rate as a function of crack extension, is obtained without optical determination of crack tip position. Crack length and energy release rate are determined at each point of the test based on experimental compliance until unstable advance occurs. In order to confirm the theoretical models, unidirectional carbon/epoxy specimens have been tested. Experimental data are evaluated by means of two reduction schemes: an existing data method named Corrected Beam Theory with effective crack length and the new method named Beam Theory including Bending Rotation effects. Shear and local deformation effects are included in both reduction schemes. Results concerning the determination of crack length without crack advance and during stable crack propagation are presented. The agreement between experimental values and theoretical results obtained by the new approach is excellent. Based on the accurate crack length determination at each point of the test, energy release rate is determined point to point and therefore R-curve is obtained.

Shear moduli for non-isotropic, open cell foams using a general elongated Kelvin foam model

Equations for calculating the shear modulus of non-isotropic, open cell foams in the plane perpendicular to the rise direction and in a plane parallel to the rise direction are derived using an elongated Kelvin foam model. This Kelvin foam model is more general than that employed by previous authors as the size and shape of the unit cell are defined by specifying three independent cell dimensions. The equations for the shear compliances are derived as a function of three unit cell dimensions and the section properties of the cell edges. From the compliance equations, the shear modulus equations are obtained and written as a function of the relative density and two unit cell shape parameters. The dependence of the two shear moduli on the relative density and the two shape parameters is demonstrated.

Kinetostatic modeling of 3-RRR compliant micro-motion stages with flexure hinges

This paper presents the derivation of the kinetostatic model of a flexure-based 3-RRR compliant micro-motion stage. The kinetostatic model has closed-form equations and flexure hinge compliances are one of the variables in the model. Researchers have an option of selecting the most suitable flexure hinge compliance equations to calculate the kinetostatics of the stage. Two cases are studied where two kinetostatic results are obtained using two different sets of flexure hinge equations. The kinetostatic results are compared to the finite-element-analysis results to verify their accuracies.

A new generalized model for elliptical arc flexure hinges

Flexure hinges have been used in many engineering areas where high precision and sensitivity are required. Many kinds of flexure profiles were proposed during the past decade. Therefore, a general closed-form solution for flexure hinges of different profiles that incorporates the profile selection and parameter design will be of great benefit to the hinge design. The present work brings circular, right-circular, and elliptical profiles together by proposing a generalized flexure hinge model, which we call elliptical arc flexure hinges (whose maximum eccentric angle phi(m) ranges from 0 to pi/2) to distinguish from the existing elliptical flexure hinge (phi(m)=pi/2). Based on the theories of mechanics of materials, all the elements in the compliance matrix for elliptical arc flexure hinges are deduced by introducing the eccentric angle of ellipse as the integral variable. These compliance equations simply boil down to four integrals, thus simplifying the compliance calculation. These equations also apply to elliptical (phi(m)=pi/2), circular (a=b), and right-circular (phi(m)=pi/2 and a=b) hinges. These compliance equations were checked by comparing them with the results of finite element analysis, the existing equations, and experiment results. The comparison results show that these generalized equations are concise and adequate for most design purposes.

The effect of the accuracies of flexure hinge equations on the output compliances of planar micro-motion stages

This paper presents the effect of the accuracies of flexure hinge compliance equations (the accuracies vary with a geometrical ratio, t/R of a hinge) on the output compliances of RRR and 3-RRR micro-motion stages. Closed-form output compliance models of the stages are derived (with flexure hinge compliances as one of the variables) to investigate the aforementioned effect. The output compliances, calculated using various flexure hinge equations, are compared to that of the finite element analysis (FEA) and the results are discussed.

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

Abstract The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]6 , [±30°]5, and [±45°]5 laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.

COMPLIANCE CALCULATION OF ELLIPTICAL FLEXURE HINGE

Based on the bending theory of beam with variable cross-section in mechanics of materials, the compliance equations of elliptical flexure hinge are deduced by introducing centrifugal angle as the integral variable. By defining an intermediate parameter in the equations, more concise compliance equations of elliptical flexure hinge are presented, thus avoiding time-consuming numerical integrals. When the minor axis and the major axis are equal, the hinge becomes a right-circular flexure hinge, and these compliance equations converge to equations for right-circular flexure hinge. A number of elliptical flexure hinges with different shapes are analyzed by using finite element software ANSYS. The results of the finite element analysis and the equation computation ones are in good agreement, which indicate that the compliance equations are correct.

Design, Modeling, and Initial Experiments on Microscale Amplification Device

This article proposes the design of several microelectromechanical amplification devices formed of many identical microunits that are connected in a serial–parallel configuration, each being individually actuated and amplifying its own input motion. The microdevices realize the border-crossing from the micro-to the meso-scale displacement domain as they combine the micron-level individual inputs into millimeter-range output levels. The base structural unit is a flexure-based compliant device that is capable of transforming the input from a thermal actuator into an amplified displacement, about a direction perpendicular to the input one. The base unit is designed based on performance criteria, such as displacement amplification, input stiffness and output stiffness by utilizing finite element simulation, and an algorithm based on closed-form compliance equations of the incorporated flexure hinges. The microelectromechanical amplification device is monitored by means of embedded capacitive displacement sensors for the input port. The feasibility of the device design was verified through a numerical simulation and some initial experimental results are presented.

Mode-II Fracture in E-glass-polyester Composite

In this work a novel beam analysis of the end-loaded split (ELS) and over-notched flexure (ONF) specimens are presented. New compliance and strain energy release rate equations are developed incorporating crack tip deformation and transverse shear analysis. The Saint Venant effect at the clamped end of the ELS specimen is also considered. Unidirectional glass-polyester specimens are manufactured and experimental (initiation and propagation) tests are performed to confirm the theoretical models. The experimental data is evaluated by means of four reduction techniques: the exact beam theory, virtual crack closure technique, compliance calibration method, and direct beam theory. It is shown that the results by the four methods agree quite closely; however, the compliance calibration method is not recommended for initiation data reduction in the case of the ONF specimen. The ELS specimens suffer from relatively large displacements. Thus, propagation tests are not possible to be performed. The ONF test seems to be a better tool to evaluate the mode-II toughness of composite materials with low flexural modulus.