Free-edge stress concentration in angle-ply laminates

Summary For a symmetric [±]s-angle-ply laminate coupon within the framework of linear elasticity the free-edge effect is treated in a closed-form analytical way. To that end, for uniaxial tension the near-edge deformation is traced back to a complex-analytical potential. For the given boundary conditions the respective potential is determined and thus the resultant displacement field and the accompanying stresses. Thereby, in particular, a logarithmic singularity is obtained for the free-edge stress concentration.     

Buckling of non-uniformly distributed graphene and fibre reinforced multiscale angle-ply laminates

Meccanica - Present work investigates the biaxial buckling of three-phase, angle-ply laminates reinforced with graphene platelets and carbon or glass fibres. The analysis is based on Classical...     

Effect of transverse cracks on the mechanical properties of angle-ply composites laminates

A modified shear lag analysis, taking into account the notion of stress perturbation function, is employed to evaluate the effect of transverse cracks on the stiffness reduction in [±θ_n/90_m]_S angle-ply laminated composites. Effects of number of 90° layers and number of ±θ layers on the laminate stiffness have also been studied. The present results represent well the dependence of the degradation of mechanical properties on the fibre orientation angle of the outer layers, the number of cracked cross-ply layers and the number of uncracked outer ±θ layers in the laminate.     

Photoviscoelastic stress analysis of a plate with a central hole

An epoxy resin containing excessive plasticizer was developed and characterized. The material, which deforms viscously at room temperature, has optical properties that depend on stress and strain. A tensile specimen was prepared from the epoxy resin so that the mechanical and optical properties of the epoxy resin could be characterized. The elastic and plastic behavior was determined at 37°C using tensile stresses between 4 and 26 MPa. The birefringence was also recorded as a function of time and stress. From these results, a photoviscoelastic constitutive equation was constructed to describe the dependence of the birefringence on stress and strain. The constitutive equation was then applied to study the deformation of a tensile specimen containing a central circular hole. By using the isochromatic fringes in combination with the isoclinic, the time-dependent variation of the stress field in the specimen was solved.     

Singular stress fields of angle-ply and monoclinic bimaterial wedges

The characteristic equations for the order of stress singularity of anisotropic bimaterial wedges subjected to traction boundary conditions are investigated. For an angle-ply bimaterial wedge, both fully bonded and frictional interfaces are considered, whereas for a monoclinic bimaterial wedge, a frictional interface is considered. Here, the Stroh formalism and the separation of variables technique are used. In general, the order of stress singularity can be real or complex, but for the special geometry of a crack along the frictional interface of a monoclinic composite, it is always real. Explicit characteristic equations for the order of singularity are presented for an aligned orthotropic composite with a frictional interface. Numerical results are given for an angle-ply bimaterial wedge and a monoclinic bimaterial wedge consisting of a graphite/epoxy fiber-reinforced composite.     

Effect of transverse cracks on the effective thermal expansion coefficient of aged angle-ply composites laminates

A modified shear-lag analysis, taking into account the concept of stress perturbation function, is developed and applied to evaluate the effect of transverse cracks on the effective thermal expansion coefficient of aged angle-ply composites laminates. Effects of number of 90° layers and number of θ° layers in the outer angle-ply laminates on the reduction of the effective axial coefficient of thermal expansion have also been studied. The results of this paper represent well the dependence of the reduction of the effective axial coefficient of thermal expansion on the hygrothermal conditions, the fibre orientation angle of the outer layers, the number of cracked cross-ply layers and the number of un-cracked outer θ° layers in laminate.     

A twofold strength and toughness criterion for the onset of free-edge shear delamination in angle-ply laminates

Free edge delamination in composite structures results from very localised stress fields which induce a stress concentration promoting the nucleation of an interfacial crack. To predict such a delamination onset at the free edge of a $(\pm \theta {)}_s$ laminate in traction, use is made of a strength and toughness criterion which combines a stress condition with an energy analysis. A generalised plane strain model allows to determine the stress distribution near the free edge and the energy released by the nucleation of an interfacial crack. The results show that this approach can predict the delamination onset for $((\pm 10{)}_s\text{,}(\pm 20{)}_s)$ laminates provided the interfacial fracture energy and interlaminar shear strength are known. These characteristic values can be identified with the help of traction tests performed on samples with different thicknesses.     

A variational model for stress analysis in cracked laminates with arbitrary symmetric lay-up under general in-plane loading

The present research work presents a variational approach for stress analysis in a general symmetric laminate, having a uniform distribution of ply cracks in a single orientation, subject to general in-plane loading. Using the principle of minimum complementary energy, an optimal admissible stress field is derived that satisfies equilibrium, boundary and traction continuity conditions. Natural boundary conditions have been derived from the variational principle to overcome the limitations of the existing methodology on the analysis of general symmetric laminates. Thus, a systematic way to formulate boundary value problem for general symmetric laminates containing many cracked and un-cracked plies has been derived, and appropriate mathematical tools can then be employed to solve them. The obtained results are in excellent agreement with the available results in the literature. In the field of matrix cracks analysis for symmetric laminates, the present formulation is the most complete variational model developed so far.     

Dynamic stress analysis of a functionally graded material plate with a circular hole

This paper is to study the two-dimensional dynamic stress of a functionally graded material (FGM) plate with a circular hole under plane compressional waves at infinity. With using the method of piece-wise homogeneous layers, the dynamic stress distribution of the FGM plate having radial arbitrary material parameters is derived based on the complex variable method. As examples, numerical results are presented for the FGM plate having given radial shear modulus, density and Poisson’s ratio. It is found that the dynamic stress around the circular hole in the FGM plate can be effectively reduced by choosing the proper change ways of the radial material parameters for different frequency incident wave.     

A new Kirchhoff plate model based on a modified couple stress theory

In this paper a new Kirchhoff plate model is developed for the static analysis of isotropic micro-plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a detailed variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Method of Fundamental Solutions (MFS) which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new Kirchhoff plate model.     

Free-edge interlaminar stress analysis of piezo-bonded composite laminates under symmetric electric excitation

A stress function-based approach is proposed to analyze the free-edge interlaminar stresses of piezo-bonded symmetric laminates. The proposed method satisfies the traction free boundary conditions, as well as surface free conditions. The symmetric laminated structure was excited under electric fields that can generate induced strain, resulting in pure extension in the laminated plate. The governing equations were obtained by taking the principle of complementary virtual work. To verify the proposed method, cross-ply, angle-ply and quasi-isotropic laminates were analyzed. The stress concentrations predicted by the present method were compared with those analyzed by the finite element method. The results show that the stress function-based analysis of piezo-bonded laminated composite structures is an efficient and accurate method for the initial design stage of piezo-bonded composite structures.     

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates

A Reissner–Mindlin theory for composite laminates without invoking ad hoc kinematic assumptions is constructed using the variational-asymptotic method. Instead of assuming a priori the distribution of three-dimensional displacements in terms of two-dimensional plate displacements as what is usually done in typical plate theories, an exact intrinsic formulation has been achieved by introducing unknown three-dimensional warping functions. Then the variational-asymptotic method is applied to systematically decouple the original three-dimensional problem into a one-dimensional through-the-thickness analysis and a two-dimensional plate analysis. The resulting theory is an equivalent single-layer Reissner–Mindlin theory with an excellent accuracy comparable to that of higher-order, layer-wise theories. The present work is extended from the previous theory developed by the writer and his co-workers with several sizable contributions: (a) six more constants (33 in total) are introduced to allow maximum freedom to transform the asymptotically correct energy into a Reissner–Mindlin model; (b) the semi-definite programming technique is used to seek the optimum Reissner–Mindlin model. Furthermore, it is proved the first time that the recovered three-dimensional quantities exactly satisfy the continuity conditions on the interface between different layers and traction boundary conditions on the bottom and top surfaces. It is also shown that two of the equilibrium equations of three-dimensional elasticity can be satisfied asymptotically, and the third one can be satisfied approximately so that the difference between the Reissner–Mindlin model and the second-order asymptotical model can be minimized. Numerical examples are presented to compare with the exact solution as well as the classical lamination theory and the first-order shear-deformation theory, demonstrating that the present theory has an excellent agreement with the exact solution.     

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.     

3D stress local nondestructive analysis in the middle plane of a plate

The Isodyne technique based on the scattered light scanning from a thin laser beam can be used to obtain the information inside the loaded object in the room temperature nondestructively, so it is a very powerful technique in 3D stress analysis. The problems are how to interpret the information and how to get sufficient information from the few interference fringe. Birefringence phase shift technique can distinguish the fringe orders automatically and enrich the information in 256 gray levels between maxim and minim light intensity. In the paper the Isodyne birefringence phase shift method with an oblique incidence and equilibrium equation is presented, by which the 3D stresses in the middle plane of a plate with U shape notch are separated successfully. The project supported by the National Natural Science Foundation of China under the Grant No. 1380345     

Velocity dispersion in an elastic plate with microstructure: effects of characteristic length in a couple stress model

Microstructural effects become important, when dimensions of the heterogeneous material are comparable to the length scale of microstructure and the state of stress needs to be defined in a non-local manner. Linear theory of elasticity, which is associated with the concept of homogeneity of material and local stresses, cannot describe the behavior of the materials with microstructures. In this study, Couple stress theory of elasticity has been employed to capture the size effects on the propagation of Lamb waves in an elastic plate with microstructure. Effects on the dispersion curves of Lamb waves are studied, when the characteristic length of the material is comparable to cell size. The governing equations of couple stress theory, involving stresses and couple stresses are solved to study the impact of different characteristic lengths, comparable with cell size. Since bone is a material with microstructure, so for numerical calculations and graphical representation of the results, the plate is considered to have mechanical properties typically used for bones.     

Creep analysis with a stress range dependent constitutive model

Many materials exhibit the stress range dependent creep behavior. The power law creep observed for a certain stress range changes to the viscous type creep as the stress value decreases. Recently published experimental data for advanced heat resistant steels indicate that the high creep exponent (in the range 7–12) may decrease to the low value of approximately unity within the stress range relevant for engineering structures. The aim of this paper is to analyze the influence of the stress range dependent power-law-viscous creep transition on the behavior of structures at elevated temperature. A constitutive model for the minimum creep rate is introduced to describe both the linear and the power law creep depending upon the stress level. To demonstrate basic features of the stress range dependent creep modeling, several elementary examples from structural mechanics are presented. They include a stress relaxation problem, a beam subjected to pure bending and a pressurized thick-walled cylinder. Based on the uni-axial transition stress the transition value of the external load is estimated such that above this value the power law can be applied. For the loading levels below this value the character of the stress distribution as well as the stress values are essentially influenced by the viscous creep.     

Theoretical basis and general optimal formulations of isoparametric generalized hybrid/mixed element model for improved stress analysis

By the modified three-field Hu-Washizu principle, this paper establishes a theoretical foundation and general convenient formulations to generate convergent stable generalized hybrid/mixed element (GH/ME) model which is invariant with respect to coordinate, insensitive to geometric distortion and suitable for improved stress computation. In the two proposed formulations, the stress equilibrium and orthogonality constraints are imposed through incompatible displacement and internal strain modes respectively. The proposed model by the general formulations in this paper is characterized by including assumed stress/strain, assumed stress, variable-node, singular, compatible and incompatible GH/ME models. When using regular meshes or the constant values of the isoparametric Jacobian Det in the assumed strain interpolation, the incompatible GH/ME model degenerates to the hybrid/mixed element model. Both general and concrete guidelines for the optimal selection of element shape functions are suggested. By means of the GH/ME theory in this paper, a family of new GH/ME can be and have been easily constructed. The software can also be developed conveniently because all the standard subroutines for the corresponding isoparametric displacement elements can be utilized directly. Modified version of a conference paper presented at Int. Conf. on EPMESC IV, July 29–Aug. 3, 1992, Dalian, China     

Analysis of complex stress intensities for cracked laminates

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