Transverse shear stress distribution through thickness near an internal part-through elliptical hole in a stretched plate

A semi-analytical post-processing method, termed the equilibrium/compatibility method here, is used for computation of hitherto unavailable through-thickness variation of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of an internal part-through elliptical cylindrical hole weakening an edge-loaded rectangular plate. A C⁰-type triangular “composite” plate element, based on the assumptions of transverse inextensibility and piece (“layer”)-wise constant shear-angle theory (LCST), is employed to first compute the inplane stresses and “layer”-wise through-thickness average transverse shear stresses. These serve as the starting point for computation of through-thickness distribution of transverse shear stresses in the vicinity of the circumferential re-entrant corner line of the internal part-through elliptical hole. As in the case of its circular counterpart, the transverse shear stresses computed by the conventional equilibrium method (EM) are, in contrast, in serious error in the presence of the circumferential re-entrant corner line singularity arising out of the internal part-through elliptical hole, and are found to violate the compatibility condition. The computed maximum transverse shear stress can be high enough to cause catastrophic transverse shear fracture in the shape of a cone, of elliptical cross-section starting from the circumferential re-entrant corner line of the internal part-through hole. The results computed by the present analysis are in line with a three-dimensional asymptotic analysis.     

Finite element analysis of free-edge stresses in composite laminates under mechanical an thermal loading

In this paper, a multi-particle finite element [Nguyen VT, Caron JF. A new finite element for free edge effect analysis in laminated composites. Comput Struct, accepted for publication] is applied for general laminated and is shown to be capable of simultaneously predicting global and local responses. The analysis of free-edge stresses of composite laminates subjected to mechanical and thermal loads is performed using this C^o${\mathbb{C}}^o$ eight-node layer-wise finite element after a classical bending validation. Laminates with finite dimensions are considered and three-dimensional out-of-plane stresses in the interior and near the free edges are evaluated. The results obtained with this finite element modelling are compared with those available in the literature. The present calculation provides accurate stresses and can be utilised as and operational tool to predict interlaminar stresses under the loads of mechanical and thermal combined.     

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical $\text{1}\text{√r}$ stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.     

Three-dimensional asymptotic stress field in the vicinity of the line of intersection of a circular cylindrical through/part–through open/rigidly plugged hole and a plate

Heretofore unavailable asymptotic solutions to a class of problems pertaining to the stress fields in the neighborhood of the circumferential corner line or line of intersection of a circular cylindrical through or part-through (embedded) open or rigidly plugged hole and a bounding or interior surface of an isotropic plate, subjected to far-field extension-bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, are presented. A local orthogonal curvilinear coordinate system (ρ,φ,θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential corner line. One of the components of the Euclidean metric tensor, namely g₃₃, is approximated (ρ/a?1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. Nine different combinations of boundary conditions are considered, five of which relate to a through hole or infinitely rigid inclusion, while the remaining four pertain to a part–through (embedded) hole or infinitely rigid inclusion. Numerical results presented include the effect of Poisson's ratio, wherever applicable, on the computed lowest eigenvalue(s).     

Debonding of rigid curvilinear inclusions in longitudinal shear deformation

The problem of two symmetrically placed interface cracks at rigid curvilinear inclusions under longitudinal shear deformation is considered. A solution valid for arbitrary inclusion shapes is found. It depends on a parameter β describing the cracks. For $\text{β = e}{}^{\mathrm{i\alpha }}$ where α is an angle, the cracks lie in the interface. For β real and greater than unity, we have two radial cracks emanating from a curvilinear cavity. The solution for $\text{β = 1}$ corresponds to a completely debonded inclusion.Examples of elliptic, square with rounded corners, and rectangular inclusions are worked out in detail. It is shown that the crack tip stress intensity factor becomes infinite for interface cracks terminating at cusps and corners. This phenomenon is attributed to the change in the nature of the singularity as the crack tip approaches a cusp or corner. The singularity is three-quarter power at a cusp and two-thirds power at a corner of a rectangular inclusion. Finally, the application of the results to composite materials is indicated.     

铆接界面端三维奇异性应力场的研究

针对铆接结构的特点,应用特征函数扩展技术分析柱坐标下接触界面端的应力奇异性问题。建立了柱坐标下圆柱体端面接触边缘附近的三维渐近位移场和应力场渐近表达式,并根据铆钉/被铆接件接触界面端的位移和应力边界条件,建立一个非线性特征方程组。据此方程组可求解界面端邻域的应力奇异性指数、位移和应力角分布函数的数值解。通过与有限元方法计算结果相对比,验证了该方法的有效性。分析了平头、沉头以及半圆头铆钉构成的铆接结构的应力奇异性问题,考察了铆钉材料、几何形式和摩擦系数对接触界面端应力奇异性指数和应力场角分布的影响。     

On three-dimensional singular stress field at the front of a planar rigid inclusion (anticrack) in an orthorhombic mono-crystalline plate

A novel eigenfunction expansion technique, based in part on separation of the thickness-variable and partly on the Eshelby–Stroh type affine transformation, is developed to derive three-dimensional asymptotic stress field in the vicinity of the front of a semi-infinite through-thickness anticrack reinforcing an infinite orthorhombic single crystal plate, of finite thickness and subjected to far-field mode I/II loadings. Anticrack-face boundary conditions and those that are prescribed on the top and bottom (free or fixed) surfaces of the plate are exactly satisfied. The present investigation considers six through-anticrack systems reinforcing orthorhombic single crystal plates. Explicit expressions for the singular stresses in the vicinity of the front of a through-thickness anticrack reinforcing an orthorhombic plate, subjected to far-field mode I/II loadings, are presented. Finally, hitherto largely unavailable results, pertaining to the through-thickness variations of stress singularity coefficients corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the top and bottom surfaces of an orthorhombic mono-crystalline plate under investigation, bridge a longstanding gap in the stress singularity/fracture mechanics literature.     

Interlaminar stress recovery for three-dimensional finite elements

An accurate evaluation of interlaminar stresses in multilayer composite laminates is crucial for the correct prediction of the onset of delamination. In general, three-dimensional finite element models are required for acceptable accuracy, especially near free edges and stress concentrations. Interlaminar stresses are continuous both across and along layer interfaces. Nonetheless, the continuity of interlaminar stresses is difficult to enforce in C⁰$C^0$ interpolated elements. Nodal values of the stresses are usually retrieved using extrapolation techniques from super-convergent points, if known, or Gauss points inside the element. Stress fields within an element can be deduced using either constitutive relations or variationally consistent procedures. In either case, spurious oscillations in stress fields may be encountered leading to a reduced accuracy of the recovered stresses at nodes. In this paper, an efficient interlaminar stress recovery procedure for three-dimensional finite element formulations is presented. The proposed procedure does not rely on extrapolation techniques from super-convergent or integration points. Interlaminar stress values are retrieved directly at nodes and stress continuity at the inter-element boundary is automatically satisfied. Several benchmark problems were analysed. Comparisons with finite element software and available solutions in the literature confirmed the accuracy of the procedure. Accurate interlaminar stresses were obtained using coarser meshes compared to customary recovery procedures.     

Elastodynamic near-tip fields for a rapidly propagating interface crack

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar isotropic elastic solids is investigated. Both anti-plane and in-plane motions are considered. The anti-plane displacements and the in-plane displacement potentials are sought in the separated forms $\text{r}{}^{\mathrm{q}}\text{F(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. The mathematical statement of the problem reduces to a second-order linear ordinary differential equation in θ, which can be solved analytically. Formulation of the boundary and interface conditions leads to an eigenvalue problem for the singularity exponent $\text{q}$. For the in-plane problem, root $\text{q}$ is found to be complex. Thus, the stresses exhibit violent oscillations within a small region around the crack tip, and the solutions have physical significance only outside this region. The angular stress distributions are plotted for various crack speeds, and it is found that at a high enough speeds the direction θ of maximum stress moves out of the interface. This result indicates that a running interface crack may move into one of the adjoining materials.     

Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading

The interlaminar stresses and displacements near the free-edges and ply cracks are investigated by using the state space equation method for general cross-ply laminates subjected to extension and/or thermal loading. By this approach, a laminated plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. Numerical solutions are compared with results obtained from other methods. It is found that the theory provides a satisfactory approximation to the stress singularity occurring in the vicinity of the free-edges and ply cracks.     

Stress singularities in bimaterial bodies of revolution

An eigenfunction expansion solution is first developed for determining the stress singularities of bimaterial bodies of revolution by directly solving the equilibrium equations of three-dimensional elasticity in terms of displacement functions. The characteristic equations are explicitly given for determining the stress singularities in the vicinity of the interface corner of two intersecting bodies of revolution having a sharp corner with free boundary conditions along the corner. The characteristic equations are found to be equivalent to a combination of the characteristic equations for plane elasticity problems and St. Venant torsion problems. The strength of stress singularities varying with the interface angles is also investigated. The first known asymptotic solutions for the displacement and stress fields are also explicitly shown for some interface angles. The present results will be useful not only for understanding the singularity behaviors of stresses in the vicinity of a revolution interface corner, but also for developing accurate numerical solutions with fast convergence for stress or vibration analysis of a body of revolution having an interface corner.     

Identifying through-thickness material properties of carbon-fiber-reinforced plastics using the virtual fields method combined with moiré interferometry

An investigation into the use of a novel curved-beam composite specimen is conducted to measure the interlaminar (through-thickness) tensile properties of carbon-fiber-reinforced plastic. A combination of a numerical model and full-field displacement/strain measurement with moiré interferometry is utilized in this study. Through-thickness material properties are identified from the measured displacement distribution using the virtual fields method. Because of the shape and the loading condition of the proposed curved composite beam, both tensile and shear stresses exist in the through-the-thickness direction. Therefore, the interlaminar tensile modulus, as well as the interlaminar shear modulus, can be evaluated. The measurement results by moiré interferometry provide the material properties through inverse analysis.     

开孔碳纤维层合板层间应力分析

针对具有典型铺设角 的开孔碳纤维层合板, 采用三维有限元数值模拟方法, 分析了在单向拉伸载荷作用下孔边附近的层间应力, 讨论了界面层参数对层间应力的影响, 详细给出了典型铺设角之间层间应力的分布规律和最大层间应力产生的位置。结果表明: 对于相同铺设角的界面层, 沿厚度方向的位置影响层间应力的大小, 但不影响分布趋势; 而铺层顺序(如 或 )对层间应力的大小和分布趋势影响则较小。最大层间正应力产生于 的界面层, 位于与拉伸方向成90°的位置, 是外加拉伸应力的51%; 最大层间剪应力产生于 的界面层, 最大层间环向剪应力位于与拉伸方向成74°的位置, 是外加拉伸应力的64%; 而最大层间径向剪应力位于与拉伸方向成66°的位置, 是外加拉伸应力的25%。      

Experimental determination of residual stresses in composite laminates [02/θ2]s

The present work aims to determine the residual stresses in carbon fiber/epoxy composite laminates, by means of the incremental hole-drilling method. Based on mechanical theories of composite laminates and an elastic plate with a circular hole, the relationship between the relaxed strains on the surface of laminates and the residual stresses in laminates was established. This newly deduced theoretical formula was adapted into the incremental hole-drilling method and allowed us to further study the residual stresses in the through-thickness direction for various composite laminates. Related numerical modeling of composite laminate with a hole was built to calibrate the coefficients within the formula. Experiments were conducted and the residual stresses in composite laminates [0₂/θ₂]_s are presented. The proposed approach was validated with the consistence between our results for cross-ply laminates and those in literature.     

Numerical analysis on singular solutions of the Poisson equation in two-dimensions

 A numerical method for extracting the coefficients of the asymptotic series solution of the Poisson equation in two dimensions in the vicinity of singular points is presented. This method is an extension of that presented in (Szabó and Yosibash 1996) to non-homogeneous boundary value problems, and is general in the sense that it is applicable to almost any type of point singularity. Numerical experiments for crack-tip singularities, re-entrant corner singularities, abrupt change in boundary conditions, and singularities associated with a multi-material inclusion are presented to substantiate the proposed techniques. Constant as well as varying non-homogeneous “right-hand-side” functions are studied.     

On a hybrid method to determine strain fields around propagating cracks

The moving singularity of the crack tip in a plane-stress plate causes a highly dynamic stress field of varying intensity with time, throughout the period of the propagation of the crack. This dynamic stress field results in a considerable change of the mechanical and optical properties of a strain-rate dependent material. An analysis of this varying dynamic stress field was presented in this paper which contradicts assumptions and simplifications introduced in a previous paper [7], referring to the same problem. For the experimental determination of the $\text{K}{}^{\mathrm{d}}{}_{\mathrm{I}}\text{-}\text{factor}$ the optical method of the dynamic caustics was utilized in combination with a high-speed camera and a comparison was sketched between the possibilities of this method and the strain-gauge method used in Ref. [7].