Authors:X. Ji; F. Zhu; P. F. He Pages: 879 - 885 Abstract: In this article, a direct stress approach based on finite element analysis to determine the stress intensity factor is improved. Firstly, by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack, we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity factor, which is nearly equal to the stress intensity factor given by the asymptotic solution. Secondly, the crack problem is solved numerically by the finite element method. Depending on the modeling capability of the software, we designed an adaptive mesh model to simulate the stress singularity. Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corresponding crack problem. Therefore, the stress intensity factor may be calculated from the stress distribution in the appropriate interval, with a high accuracy. PubDate: 2017-10-01 DOI: 10.1007/s10409-017-0640-4 Issue No:Vol. 33, No. 5 (2017)

Authors:Haian Zhou; Xiaoming Wang; Huayong Wu; Jianbing Meng Pages: 926 - 941 Abstract: The vibroacoustic response and sound absorption performance of a structure composed of multilayer plates and one rigid back wall are theoretically analyzed. In this structure, all plates are two-dimensional, microperforated, and periodically rib-stiffened. To investigate such a structural system, semianalytical models of one-layer and multilayer plate structures considering the vibration effects are first developed. Then approaches of the space harmonic method and Fourier transforms are applied to a one-layer plate, and finally the cascade connection method is utilized for a multilayer plate structure. Based on fundamental acoustic formulas, the vibroacoustic responses of microperforated stiffened plates are expressed as functions of a series of harmonic amplitudes of plate displacement, which are then solved by employing the numerical truncation method. Applying the inverse Fourier transform, wave propagation, and linear addition properties, the equations of the sound pressures and absorption coefficients for the one-layer and multilayer stiffened plates in physical space are finally derived. Using numerical examples, the effects of the most important physical parameters—for example, the perforation ratio of the plate, sound incident angles, and periodical rib spacing—on sound absorption performance are examined. Numerical results indicate that the sound absorption performance of the studied structure is effectively enhanced by the flexural vibration of the plate in water. Finally, the proposed approaches are validated by comparing the results of stiffened plates of the present work with solutions from previous studies. PubDate: 2017-10-01 DOI: 10.1007/s10409-017-0659-6 Issue No:Vol. 33, No. 5 (2017)

Authors:Guanghui Qing; Jia Tian Abstract: For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element (NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses. PubDate: 2017-11-09 DOI: 10.1007/s10409-017-0727-y

Authors:Yan Xiong; Jing Li; Zhaohui Liu; Chuguang Zheng Abstract: The absence of sub-grid scale (SGS) motions leads to severe errors in particle pair dynamics, which represents a great challenge to the large eddy simulation of particle-laden turbulent flow. In order to address this issue, data from direct numerical simulation (DNS) of homogenous isotropic turbulence coupled with Lagrangian particle tracking are used as a benchmark to evaluate the corresponding results of filtered DNS (FDNS). It is found that the filtering process in FDNS will lead to a non-monotonic variation of the particle collision statistics, including radial distribution function, radial relative velocity, and the collision kernel. The peak of radial distribution function shifts to the large-inertia region due to the lack of SGS motions, and the analysis of the local flowstructure characteristic variable at particle position indicates that the most effective interaction scale between particles and fluid eddies is increased in FDNS. Moreover, this scale shifting has an obvious effect on the odd-order moments of the probability density function of radial relative velocity, i.e. the skewness, which exhibits a strong correlation to the variance of radial distribution function in FDNS. As a whole, the radial distribution function, together with radial relative velocity, can compensate the SGS effects for the collision kernel in FDNS when the Stokes number based on the Kolmogorov time scale is greater than 3.0. However, it still leaves considerable errors for \({ St}_\mathrm{k }<3.0\) . PubDate: 2017-11-08 DOI: 10.1007/s10409-017-0720-5

Authors:Jingru Song; Cuncai Fan; Hansong Ma; Lihong Liang; Yueguang Wei Abstract: For decades, nacre has inspired researchers because of its sophisticated hierarchical structure and remarkable mechanical properties, especially its extreme fracture toughness compared with that of its predominant constituent, \(\hbox {CaCO}_{3}\) , in the form of aragonite. Crack deflection has been extensively reported and regarded as the principal toughening mechanism for nacre. In this paper, our attention is focused on crack evolution in nacre under a quasi-static state. We use the notched three-point bending test of dehydrated nacre in situ in a scanning electron microscope (SEM) to monitor the evolution of damage mechanisms ahead of the crack tip. The observations show that the crack deflection actually occurs by constrained microcracking. On the basis of our findings, a crack propagation model is proposed, which will contribute to uncovering the underlying mechanisms of nacre’s fracture toughness and its damage evolution. These investigations would be of great value to the design and synthesis of novel biomimetic materials. PubDate: 2017-11-07 DOI: 10.1007/s10409-017-0724-1

Authors:Miodrag Zigic; Nenad Grahovac Abstract: This paper deals with rigid body attitude estimation on the basis of the data obtained from an inertial measurement unit mounted on the body. The aim of this work is to present the numerical algorithm, which can be easily applied to the wide class of problems concerning rigid body positioning, arising in aerospace and marine engineering, or in increasingly popular robotic systems and unmanned aerial vehicles. Following the considerations of kinematics of rigid bodies, the relations between accelerations of different points of the body are given. A rotation matrix is formed using the quaternion approach to avoid singularities. We present numerical procedures for determination of the absolute accelerations of the center of mass and of an arbitrary point of the body expressed in the inertial reference frame, as well as its attitude. An application of the algorithm to the example of a heavy symmetrical gyroscope is presented, where input data for the numerical procedure are obtained from the solution of differential equations of motion, instead of using sensor measurements. PubDate: 2017-11-07 DOI: 10.1007/s10409-017-0726-z

Authors:L. Lan; Y. Ni; Y. Jiang; J. Li Abstract: The motion of the moonlet Dactyl in the binary system 243 Ida is investigated in this paper. First, periodic orbits in the vicinity of the primary are calculated, including the orbits around the equilibrium points and large-scale orbits. The Floquet multipliers’ topological cases of periodic orbits are calculated to study the orbits’ stabilities. During the continuation of the retrograde near-circular orbits near the equatorial plane, two period-doubling bifurcations and one Neimark–Sacker bifurcation occur one by one, leading to two stable regions and two unstable regions. Bifurcations occur at the boundaries of these regions. Periodic orbits in the stable regions are all stable, but in the unstable regions are all unstable. Moreover, many quasi-periodic orbits exist near the equatorial plane. Long-term integration indicates that a particle in a quasi-periodic orbit runs in a space like a tire. Quasi-periodic orbits in different regions have different styles of motion indicated by the Poincare sections. There is the possibility that moonlet Dactyl is in a quasi-periodic orbit near the stable region I, which is enlightening for the stability of the binary system. PubDate: 2017-11-07 DOI: 10.1007/s10409-017-0722-3

Authors:Qiang Zhang; Cheng Li; Xiaowei Quan; Yanning Wang; Liyuan Yu; Binsong Jiang Abstract: A reasonable strength criterion should reflect the hydrostatic pressure effect, minimum principal stress effect, and intermediate principal stress effect. The former two effects can be described by the meridian curves, and the last one mainly depends on the Lode angle dependence function. Among three conventional strength criteria, i.e. Mohr–Coulomb (MC), Hoek–Brown (HB), and Exponent (EP) criteria, the difference between generalized compression and extension strength of EP criterion experience a firstly increase then decrease process, and tends to be zero when hydrostatic pressure is big enough. This is in accordance with intrinsic rock strength characterization. Moreover, the critical hydrostatic pressure \(I_\mathrm{c}\) corresponding to the maximum difference of between generalized compression and extension strength can be easily adjusted by minimum principal stress influence parameter K. So, the exponent function is a more reasonable meridian curves, which well reflects the hydrostatic pressure effect and is employed to describe the generalized compression and extension strength. Meanwhile, three Lode angle dependence functions of \(L_{{\mathrm{MN}}}\) , \(L_{{\mathrm{WW}}}\) , and \(L_{{\mathrm{YMH}}}\) , which unconditionally satisfy the convexity and differential requirements, are employed to represent the intermediate principal stress effect. Realizing the actual strength surface should be located between the generalized compression and extension surface, new true-triaxial criteria are proposed by combining the two states of EP criterion by Lode angle dependence function with a same lode angle. The proposed new true-triaxial criteria have the same strength parameters as EP criterion. Finally, 14 groups of triaxial test data are employed to validate the proposed criteria. The results show that the three new true-triaxial exponent criteria, especially the Exponent Willam-Warnke criterion (EPWW) criterion, give much lower misfits, which illustrates that the EP criterion and \(L_{{\mathrm{WW}}}\) have more reasonable meridian and deviatoric function form, respectively. The proposed new true-triaxial strength criteria can provide theoretical foundation for stability analysis and optimization of support design of rock engineering. PubDate: 2017-10-31 DOI: 10.1007/s10409-017-0723-2

Authors:Xing-Quan Wang; Qing-Sheng Yang Abstract: Smart hydrogels are environmentally sensitive hydrogels, which can produce a sensitive response to external stimuli, and often exhibit the characteristics of multi filed coupling. In this paper, a hydrogel rod under chemo-mechanical coupling was analytically studied based on a poroelastical model. The already known constitutive and governing equations were simplified into the one dimensional case, then two different boundary conditions were considered. The expressions of concentration, displacement, chemical potential and stress related to time were obtained in a series form. Examples illustrate the interaction mechanism of chemical and mechanical effect. It was found that there was a balance state in the diffusion of concentration and the diffusion process could lead to the expansion or the stress change of the hydrogel rod. PubDate: 2017-10-30 DOI: 10.1007/s10409-017-0728-x

Authors:Wen-Ling Tian; Sheng-Qi Yang; Yan-Hua Huang Abstract: In this paper, a cluster model in particle flow code was used to simulate granite specimens after heat treatment under uniaxial compression. The results demonstrated that micro-cracks are randomly distributed in the specimen when the temperature is below 300 \({^{\circ }}\) C, and have partial coalescence when the temperature is up to 450 \({^{\circ }}\) C, then form macro-cracks when the temperature is above 600 \({^{\circ }}\) C. There is more inter-granular cracking than intra-granular cracking, and their ratio increases with increasing temperature. The micro-cracks are almost constant when the temperature decreases from 900 \({^{\circ }}\) C to room temperature, except for quartz \(\alpha \) – \(\beta \) phase transition temperature (573 \({^{\circ }}\) C). The fracture evolution process is obviously affected by these cracks, especially at 600–900 \({^{\circ }}\) C. Elevated temperature leads to easily developed displacement between the grains, and the capacity to store strain energy becomes weaker, corresponding to the plasticity of granite after heat treatment. PubDate: 2017-10-26 DOI: 10.1007/s10409-017-0714-3

Authors:C. F. Du; D. G. Zhang; L. Li; G. R. Liu Abstract: We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions. PubDate: 2017-10-23 DOI: 10.1007/s10409-017-0713-4

Authors:L. Wang; T. L. Jiang; H. L. Dai Abstract: This paper deals with the three-dimensional dynamics and postbuckling behavior of flexible supported pipes conveying fluid, considering flow velocities lower and higher than the critical value at which the buckling instability occurs. In the case of low flow velocity, the pipe is stable with a straight equilibrium position and the dynamics of the system can be examined using linear theory. When the flow velocity is beyond the critical value, any motions of the pipe could be around the postbuckling configuration (non-zero equilibrium position) rather than the straight equilibrium position, so nonlinear theory is required. The nonlinear equations of perturbed motions around the postbuckling configuration are derived and solved with the aid of Galerkin discretization. It is found, for a given flow velocity, that the first-mode frequency for in-plane motions is quite different from that for out-of-plane motions. However, the second- or third-mode frequencies for in-plane motions are approximately equal to their counterparts for out-of-plane motions, keeping almost constant values with increasing flow velocity. Moreover, the orientation angle of the postbuckling configuration plane for a buckled pipe can be significantly affected by initial conditions, displaying new features which have not been observed in the same pipe system factitiously supposed to deform in a single plane. PubDate: 2017-10-19 DOI: 10.1007/s10409-017-0718-z

Authors:Peng Wang; Zhijun Zheng; Shenfei Liao; Jilin Yu Abstract: The seemingly contradictory understandings of the initial crush stress of cellular materials under dynamic loadings exist in the literature, and a comprehensive analysis of this issue is carried out with using direct information of local stress and strain. Local stress/strain calculation methods are applied to determine the initial crush stresses and the strain rates at initial crush from a cell-based finite element model of irregular honeycomb under dynamic loadings. The initial crush stress under constant-velocity compression is identical to the quasi-static one, but less than the one under direct impact, i.e. the initial crush stresses under different dynamic loadings could be very different even though there is no strain-rate effect of matrix material. A power-law relation between the initial crush stress and the strain rate is explored to describe the strain-rate effect on the initial crush stress of irregular honeycomb when the local strain rate exceeds a critical value, below which there is no strain-rate effect of irregular honeycomb. Deformation mechanisms of the initial crush behavior under dynamic loadings are also explored. The deformation modes of the initial crush region in the front of plastic compaction wave are different under different dynamic loadings. PubDate: 2017-10-19 DOI: 10.1007/s10409-017-0716-1

Authors:Jian-Long Cheng; Sheng-Qi Yang; Kui Chen; Dan Ma; Feng-Yuan Li; Li-Ming Wang Abstract: In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the \(\theta = 0{^{\circ }}\) – \(45{^{\circ }}\) specimens had obvious plastic deformation during loading, and the brittleness of the \(\theta = 60{^{\circ }}\) – \(90{^{\circ }}\) specimens gradually increased during the loading process. When the anisotropic angle \(\theta \) increased from \(0{^{\circ }}\) to \(90{^{\circ }}\) , the peak strength, peak strain, and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories: tensile fracture across the discontinuities ( \(\theta = 0{^{\circ }}\) – \(30{^{\circ }})\) , sliding failure along the discontinuities ( \(\theta = 45{^{\circ }}\) – \(75{^{\circ }})\) , and tensile-split along the discontinuities ( \(\theta = 90{^{\circ }})\) . The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the \(\theta = 0{^{\circ }}\) – \(45{^{\circ }}\) specimens and was almost the same as that of the \(\theta = 60{^{\circ }}\) – \(90{^{\circ }}\) specimens. As for the strain localization highlighted in the maximum principal strain field, the \(\theta = 0{^{\circ }}\) – \(30{^{\circ }}\) specimens appeared in the rock matrix approximately parallel to the loading direction, while in the \(\theta = 45{^{\circ }}\) – \(90{^{\circ }}\) specimens it appeared at the hard and weak rock layer interface. PubDate: 2017-10-17 DOI: 10.1007/s10409-017-0706-3

Authors:Yue Zhang; Cheng Wei; Yang Zhao; Chunlin Tan; Yongjian Liu Abstract: In the conventional absolute nodal coordinate formulation (ANCF), the model is pre-meshed, the number, distribution and type of elements are unchangeable during the simulation. In addition, the deformations of a flexible body are space-varying and time-varying, one cannot predict when, where, and how the deformations will occur. Therefore, in order to obtain a satisfactory accuracy during the whole simulation, the model is usually densely meshed, but it will result in a loss of computational efficiency. In this study, an adaptive absolute nodal coordinate formulation (AANCF) is proposed to optimize the accuracy and efficiency of flexible dynamics. The movement features of flexible bodies are analyzed, and the conventional and adaptive ANCF methods are compared. Then the adaptive computation strategy is presented. The discretization errors come from the inability of interpolation functions of individual elements to capture the complexity of the exact solution, so the mesh can be adaptively optimized by changing the element sizes or the orders of interpolation functions during dynamic computation. Important issues of AANCF, including error estimation, mesh updating, and performance of the AANCF model, are analyzed and discussed in detail. A theoretical model of a planar AANCF cable is presented, where the strategies of dividing and merging elements are discussed. Moreover, the continuity of dynamic variables is deduced, and the mean factors that affect the continuity are obtained, which is very important for the subsequent continuity optimization. The simulation results indicate that the distribution of elements varies with time and space, and the elements are denser in large-deformed domains. The AANCF model improved the computational accuracy and efficiency, but the system energy is discontinuous when the elements are merged. Therefore, a continuity-optimized AANCF model is given based on the previous continuity analysis, the results show that the accuracy and continuity of energy are further improved by the continuity-optimized AANCF model. PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0721-4

Authors:Hu Liu; Jialing Yang; Hua Liu Abstract: Taylor impact is a widely used strategy in which a flat-nosed projectile is fired onto a rigid anvil directly to determine the dynamic strength of rod specimens. Nowadays, the rigid anvil is often replaced by an output target bar to ensure the accuracy of measurement via recording strain signals in the output bar. For testing the dynamic strength of low-density materials, a low-impedance target bar, which exhibits viscoelastic characteristics is often employed. In this paper, an extended Taylor model is proposed to improve the idealization of treating the target bar as perfectly rigid material in the classic Taylor model, and the viscoelastic effect of the target bar is incorporated. The viscoelastic target bar is depicted by two elastic springs and one dashpot. Based on the plastic shock wave theory in the flat-nosed projectile associated with the viscoelastic wave analysis in the target bar, the viscoelastic effect of the target bar on the impact response of the flat-nosed projectile is investigated. The finite element simulation is also carried out to verify the theoretical model, and good agreement is found. The present theoretical model is also called the Taylor-cylinder Hopkinson impact, which provides a more accurate way to identify the dynamic material parameters. The dynamic responses of the present model are further compared with previous elastic and rigid target bar models. It is found that the viscoelastic effect of the target bar should be taken into consideration in the Taylor-cylinder Hopkinson impact test for low-impedance materials. PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0717-0

Authors:Changda Wang; Xuejun Chen; Peijun Wei; Yueqiu Li Abstract: The reflection and transmission of elastic waves through a couple-stress elastic slab that is sandwiched between two couple-stress elastic half-spaces are studied in this paper. Because of the couple-stress effects, there are three types of elastic waves in the couple-stress elastic solid, two of which are dispersive. The interface conditions between two couple-stress solids involve the surface couple and rotation apart from the surface traction and displacement. The nontraditional interface conditions between the slab and two solid half-spaces are used to obtain the linear algebraic equation sets from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then, the energy fluxes carried by the various reflection and transmission waves are calculated numerically and the normal energy flux conservation is used to validate the numerical results. The special case, couple-stress elastic slab sandwiched by the classical elastic half-spaces, is also studied and compared with the situation that the classical elastic slab sandwiched by the classical elastic half-spaces. Incident longitudinal wave (P wave) and incident transverse wave (SV wave) are both considered. The influences of the couple-stress are mainly discussed based on the numerical results. It is found that the couple-stress mainly influences the transverse modes of elastic waves. PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0712-5

Authors:Wen-Feng Huang; Yu-Xin Ren; Xiong Jiang Abstract: This paper presents a simple approach for improving the performance of the weighted essentially non-oscillatory (WENO) finite volume scheme on non-uniform grids. This technique relies on the reformulation of the fifth-order WENO-JS (WENO scheme presented by Jiang and Shu in J. Comput. Phys. 126:202–228, 1995) scheme designed on uniform grids in terms of one cell-averaged value and its left and/or right interfacial values of the dependent variable. The effect of grid non-uniformity is taken into consideration by a proper interpolation of the interfacial values. On non-uniform grids, the proposed scheme is much more accurate than the original WENO-JS scheme, which was designed for uniform grids. When the grid is uniform, the resulting scheme reduces to the original WENO-JS scheme. In the meantime, the proposed scheme is computationally much more efficient than the fifth-order WENO scheme designed specifically for the non-uniform grids. A number of numerical test cases are simulated to verify the performance of the present scheme. PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0715-2

Authors:K. C. Nehar; B. E. Hachi; F. Cazes; M. Haboussi Abstract: The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors (SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method, whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials, but has to our knowledge not been used up to now for a bi-material. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency (less time consuming and less spurious boundary effect). PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0711-6

Authors:Quankun Cao; Huimin Xie Abstract: Fused deposition modelling (FDM), a widely used rapid prototyping process, is a promising technique in manufacturing engineering. In this work, a method for characterizing elastic constants of FDM-fabricated materials is proposed. First of all, according to the manufacturing process of FDM, orthotropic constitutive model is used to describe the mechanical behavior. Then the virtual fields method (VFM) is applied to characterize all the mechanical parameters \((Q_{11}\) , \(Q_{22}\) , \(Q_{12}\) , \(Q_{66})\) using the full-field strain, which is measured by digital image correlation (DIC). Since the principal axis of the FDM-fabricated structure is sometimes unknown due to the complexity of the manufacturing process, a disk in diametrical compression is used as the load configuration so that the loading angle can be changed conveniently. To verify the feasibility of the proposed method, finite element method (FEM) simulation is conducted to obtain the strain field of the disk. The simulation results show that higher accuracy can be achieved when the loading angle is close to \(30^{\circ }\) . Finally, a disk fabricated by FDM was used for the experiment. By rotating the disk, several tests with different loading angles were conducted. To determine the position of the principal axis in each test, two groups of parameters \((Q_{11}\) , \(Q_{22}\) , \(Q_{12}\) , \(Q_{66})\) are calculated by two different groups of virtual fields. Then the corresponding loading angle can be determined by minimizing the deviation between two groups of the parameters. After that, the four constants \((Q_{11}\) , \(Q_{22}\) , \(Q_{12}\) , \(Q_{66})\) were determined from the test with an angle of \(27^{\circ }\) . PubDate: 2017-10-16 DOI: 10.1007/s10409-017-0719-y