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.     

Variation-based cracked laminate analysis revisited and fundamentally extended

The analyses of cracked laminates based on a variational principle and related approaches are appraised in this paper. The limitations of the existing methodology on the analyses of more general laminate configurations have been identified. It has been revealed that the limiting factor is the lack of boundary conditions for uncracked laminae. Natural boundary conditions have then been derived from the variational principle to meet the need. Such boundary conditions are mathematically sound but cannot be simply interpreted from the physical construction of the problem intuitively. A well posed boundary value problem has thus been formulated for laminates containing however many cracked and uncracked laminae. Appropriate mathematical tools can then be employed to solve the boundary value problem. The capability of analysing cracked laminates has been enhanced significantly, as a result.     

Properties degradation induced by transverse cracks in general symmetric laminates

This paper presents the details of a methodology for predicting the thermoelastic properties degradation in general symmetric laminates with uniform ply cracks in some or all of the 90° layers. First, a stress transfer method is derived by using the concept of state space equation. The laminate can be subjected to any combination of in-plane biaxial and shear loading, and the uniform thermal loading is also taken into account. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. By this method, a laminate may be composed of an arbitrary number of monoclinic layers and each layer may have different material property and thickness. Second, the concept of the effective thermoelastic properties of a cracked laminate is introduced. Based on the numerical solutions of specially designed loading cases, the effective thermoelastic constants of a cracked laminate can be obtained. Finally, the applications of the methodology are shown by numerical examples and compared with numerical results from other models and experiment data in the literature. It is found that the theory provides good predictions of the thermoelastic properties degradation in general symmetric laminates.     

PREDICTIVE APPROACH TO FAILURE OF COMPOSITE LAMINATES WITH EQUIVALENT CONSTRAINT MODEL

<正>This work established a new analytical model based upon the equivalent constraint model(ECM)to constitute an available predictive approach for analyzing the ultimate strength and simulating the stress/strain response of general symmetric laminates subjected to combined loading,by taking into account the effect of matrix cracking.The ECM was adopted to mainly predict the in-plane stiffness reduction of the damaged laminate.Basic consideration that progressive matrix cracking provokes a re-distribution of the stress fields on each lamina within laminates, which greatly deteriorates the stress distributed in the primary load-bearing lamina and leads to the final failure of the laminates,is introduced for the construction of the failure criterion. The effects of lamina properties,lay-up configurations and loading conditions on the behaviors of the laminates were examined in this paper.A comparison of numerical results obtained from the established model and other existed models and published experimental data was presented for different material systems.The theory predictions demonstrated great match with the experimental observations investigated in this study.     

基体开裂的正交复合材料对称层合板的粘弹性力学行为

本文讨论了含有均匀基体裂纹的正交复合材料对称层合板的线性粘弹性力学行为.采用二维剪切滞后模型并对其层间剪应力在厚度方向进行线性假设分布,求得层合板的平均应力应变的线弹性解,利用等效约束模型和经典层合理论可得到层合板因为含有基体裂纹而所引起的刚度退化现象.在弹性-粘弹性对应原理的基础上对其层合板的线粘弹性进行了讨论和研究.结果表明:层合板的松弛模量和蠕变泊松比随着时间的增加而减少,到达稳态后其值基本上是恒值.并跟Zocher的解析解和有限元数值解作了比较,发现结果非常吻合.     

Variational analysis of transverse cracking and local delamination in [θm/90n]s laminates

A displacement-based variational model is developed to study the effects of transverse cracking and local delaminations in symmetric composite laminates. In the model, the crack shape is assumed to be a function of crack density and delamination length. Using a variational approach with the principle of minimum potential energy, governing equations are derived. The effective Young’s modulus E_x and energy release rate G are theoretically examined as a result of local delaminations.     

A theoretical model of matrix cracking in composite laminates under thermomechanical loading

A theoretical approach is presented for analyzing the ply cracking in general symmetric laminates subjected to any combination of in-plane mechanical loading and uniform temperature changes. The equivalent constraint model proposed by the authors in a previous work is used to account for the cracking interaction between laminae in the laminates. By using a superposition scheme and the stress field solutions the energy release rate for a ply cracking is explicitly expressed as a function of stiffness reduction parameters of the laminates. The ratio of mode I to mode II is introduced for construction of the fracture criterion. The effects of the laminate parameters and the crack spacing on the energy release rate and the mode mixity are illustrated. Finally, the model is used to predict the thermomechanical load for the first-ply-cracking. Project supported by the National Natural Science Foundation of China (No. 19972076) and the Germen Research Foundation (DFG).     

Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.     

Structure and properties of the solution space of general anisotropic laminates

The algebraic structure of the solution space of all types of anisotropic laminates is determined. The full space is shown to be the direct sum of a number of orthogonal eigenspaces, one for each simple or multiple eigenvalue, whose dimension equals the multiplicity. There are eight different types of eigenvalues, which combine to yield eleven distinct types of laminates with peculiar representations of the general solution. All such representations are explicitly obtained, along with the pseudo-metrics based on the binary product of the eigenvectors. This leads to the projection operators in the solution space, spectral sums and intrinsic tensors analogous to the Stroh–Barnett–Lothe tensors in 2-D elasticity. The present theoretical results are obtained by adopting a mixed formulation involving the deflection function and Airy’s stress function, and by using new laminate elasticity matrices different from the conventional stiffness matrices A, B and D. The new formulation also discloses an isomorphism relating each anisotropic laminate to an image laminate, such that every equilibrium solution of the former directly yields an image solution of the latter by interchanging the kinematical and kinetic variables and the in-plane and out-of-plane variables. This implies, in particular, that the classical bending theory of homogeneous plates and symmetric laminates is not a distinct subject, despite its historical development and pedagogical recognition, but is mathematically identical to the plane stress problem of anisotropic elasticity.     

An experimental and analytical treatment of matrix cracking in cross-ply laminates

The development of damage in cross-ply Hercules AS4/3502 graphite/epoxy laminates has been investigated. Specific endeavors were to identify the mechanisms for initiation and growth of matrix cracks and to determine the effect of matrix cracking on the stiffness loss in cross-ply laminates. Two types of matrix cracks were identified. These include both straight and curved cracks. The experimental study of matrix crack damage revealed that the curved cracks formed after the straight cracks and followed a repeatable pattern of location and orientation relative to the straight cracks. Therefore, it was postulated that the growth mechanism for curved cracks is driven by the stress state resulting from the formation of the straight cracks. This phenomenon was analytically investigated by a finite-element model of straight cracks in a cross-ply laminate. The finite-element results provide supporting evidence for the postulated growth mechanism. The experimental study also revealed that the number of curved cracks increased with the number of consecutive 90-deg plies. Finally, experimental results show as much as 10-percent degradation in axial stiffness due to matrix cracking in cross-ply graphite/epoxy laminates.     

基于赫林格?赖斯纳变分原理的一致高效无网格本质边界条件施加方法

无网格法具有高阶连续光滑的形函数, 在结构分析中呈现出显著的精度优势. 但无网格形函数在节点处一般没有插值性, 导致伽辽金无网格法难以直接施加本质边界条件. 采用变分一致尼兹法施加边界条件的数值解具有良好的收敛性和稳定性, 因而得到了非常广泛的应用, 然而该方法仍然需要引入人工参数来保证算法的稳定性. 本文以赫林格?赖斯纳变分原理为基础, 建立了一种变分一致的本质边界条件施加方法. 该方法采用混合离散近似赫林格?赖斯纳变分原理弱形式中的位移和应力, 其中位移采用传统无网格形函数进行离散, 而应力则在背景积分单元中近似为相应阶次的多项式. 此时的无网格离散方程可视为一种新型的尼兹法施加本质边界条件, 其中修正变分项采用再生光滑梯度和无网格形函数进行混合离散, 稳定项则内嵌于赫林格?赖斯纳变分原理弱形式中, 无需额外增加稳定项, 消除了对人工参数的依赖性. 该方法无需计算复杂耗时的形函数导数, 并满足积分约束条件, 保证了数值求解的精度. 数值结果表明, 所提方法能够保证伽辽金无网格法的计算精度最优误差收敛率, 与传统的尼兹法相比明显提高了计算效率.      

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.     

Cohesive modeling of transverse cracking in laminates under in-plane loading with a single layer of elements per ply

This study aims to bridge the gap between classical understanding of transverse cracking in cross-ply laminates and recent computational methods for the modeling of progressive laminate failure. Specifically, the study investigates under what conditions a finite element model with cohesive X-FEM cracks can reproduce the in situ effect for the ply strength. It is shown that it is possible to do so with a single element across the thickness of the ply, provided that the interface stiffness is properly selected. The optimal value for this interface stiffness is derived with an analytical shear lag model. It is also shown that, when the appropriate statistical variation of properties has been applied, models with a single element through the thickness of a ply can predict the density of transverse matrix cracks.     

Modeling of diffusion in the presence of damage in polymer matrix composites

It is now well known that Fick’s Law is frequently inadequate for describing moisture diffusion in polymers and polymer composites. Non-Fickian or anomalous diffusion is likely to occur when a polymer composite laminate is subjected to external stresses that could give rise to internal damage in the form of matrix cracks. As a result, it is necessary to take into account the combined effects of temperature, stress, and damage in the construction of such a model. In this article, a modeling methodology based on irreversible thermodynamics applied within the framework of composite macro-mechanics is presented, that would allow characterization of non-Fickian diffusion coefficients from moisture-weight-gain data for laminated composites. A symmetric damage tensor based on continuum damage mechanics is incorporated in this model by invoking the principle of invariance with respect to coordinate transformations. For tractability, the diffusion-governing equations are simplified for the special case of a laminate, with uniformly distributed matrix cracks, that is subjected to a uniaxial tensile stress. The final form for effective diffusivity obtained from this derivation indicates that the effective diffusivity for this case is a quadratic function of crack density. A finite element procedure that extends this methodology to more complex shapes and boundary conditions is also presented. Comparisons with test data for a 5-harness satin textile composite are provided for model verifications.     

复合材料刚度性能表征的协同损伤力学模型

针对复合材料层合板的弥散型损伤,提出一个刚度性能表征的协同损伤力学模型. 该模型兼顾了微观物理损伤响应和宏观材料刚度性能表征. 从微观角度,建立细观RVE 模型求解裂纹表面张开位移和滑开位移,以此定义损伤张量,并在宏观上通过对材料应变和损伤表面位移进行均匀化处理,建立单向板或层合板的损伤刚度矩阵和损伤张量之间的联系. 以基体裂纹为例,详细分析并建立了横向裂纹和纵向裂纹的损伤本构. 计算了[±θ/90₄]_S 铺层层合板中基体横向裂纹对刚度性能的影响,结果表明该方法能够准确地预测复合材料层合板由损伤导致的刚度性能衰减.      

层板复合材料的疲劳剩余刚度统计分布模型

在分析现有关于层板复合材料疲劳剩余刚度衰退模型的基础上，根据层板复合材料在疲劳载荷作用下刚度衰退变化的现象，研究了疲劳载荷对层板复合材料刚度衰退的影响，建立了一个新的、更为合理的用于描述层板复合材料在常幅疲劳载荷作用下的刚度衰退模型，导出了剩余刚度统计分布的表达式，给出了确定模型参数的方法。为验证该模型，设计了几组测试实验，并利用试验结果对模型参数进行了估计。利用该统计分布模型，可以预报层板复合材料在给定应力水平的疲劳载荷作用下循环指定周次时剩余刚度的统计分布。实验数据表明，理论预报和实验结果符合得是很好的。     

In-plane/out-of-plane concentrated forces and moments on composite laminates with elliptical elastic inclusions

The problems of composite laminates containing elliptical elastic inclusions subjected to concentrated forces and moments are considered in this paper. By employing Stroh-like formalism for the coupled stretching–bending analysis, analytical closed form solutions are obtained explicitly. The generality of the solutions provided in this paper can be shown as follows: (1) The laminates include any kinds of laminate lay-ups, symmetric or unsymmetric, which allow the stretching and bending deformations couple each other. (2) The concentrated forces and moments can be applied in in-plane and/or out-of-plane directions, located inside and/or outside the inclusions. (3) The elliptical elastic inclusions can be any kinds of elastic materials including the limiting cases such as holes, rigid inclusions, cracks, line inclusions, etc. Since no such general solution has been found in the literature, the solutions are checked and verified by the special cases that no inclusions are embedded in the laminates, and that the inclusions are replaced by holes. Moreover, with various hardness ratios of inclusion and matrix some numerical examples showing the stress resultants along the interface are presented. Like the Green’s functions for the infinite laminates and those containing holes/cracks, the present solutions associated with the in-plane concentrated forces and out-of-plane concentrated moments have exactly the same mathematical form as those of the corresponding two-dimensional problems, in which the only difference is the contents of the symbols. While for the other loading cases, new types of solutions are obtained explicitly.     

Analysis of progressive fiber debonding in elastic laminates

The evolution of fiber debonding, and sliding, in fibrous laminates is modeled by a coupled micro/macro-mechanical analysis scheme. The laminates under consideration have a symmetric layup, and are subjected to mechanical loads. The individual plies are elastic, have a unidirectional reinforcement, and can suffer local damage at the fiber/matrix interface when the resolved normal and shear stresses exceed their ultimate magnitudes. The local fields in the plies are assumed to be periodic, and are approximated by the finite element method for overall loads and local resolved stresses that are in excess of the interface strength. Local effects in the individual plies are scaled up to the laminate analysis through stress transformation factors, which are a function of the elastic properties of the plies and their stacking configuration.The proposed analysis was implemented for a periodic array model of the laminas, and for in-plane loading of the laminate. The model predictions for a unidirectional steel/epoxy system subjected to transverse loading compare remarkably well with experimental measurements. This result, and several other examples given for axial and off-axis loading of SiC/CAS laminates, illustrate the model capabilities in predicting the overall strains in the presence of simultaneous, progressive debonding in the individual plies.     

PROPERTY DEGRADATION OF ANISOTROPIC COMPOSITE LAMINATES WITH MATRIX CRACKING (I)-CONSTITUTIVE RELATION DEVELOPING FOR (θ_m/90_n)_s CRACKED LAMINATES BY STIFFNESS PARTITION

The study on property degradation of damaged composite laminates is extendedto anisotropic laminates with matrix cracking. In (I) of the paper, an idea of "stiffnesspatition" is proposed to deal with the puzzle that the in-plane normal response iscoupled with the shear response of the laminates. For (θ_m/90_n.), laminates containingtransversely cracked layers under general in-plane loading, the constitutive relationsare derived and the effective stiffnesses are expressed as the function of crack density.