Principle Of Minimum Complementary Energy Research Articles

Prediction of Tensile Modulus of 3D Four-Directional Braided Composites with Transverse Cracks

To a laminate model of 3D four-directional braided composites with periodic transverse cracks, a new normal stress function in weave direction is present. In this function, variations of stress with x and y are considered. By using the principle of minimum complementary energy, stress components in the representative unit are obtained. Tensile modulus of 3D four-directional braided composites with periodic transverse cracks is calculated by utilizing Hashin’s average theory and the principle of minimum complementary energy. To demonstrate the solution, some examples are analyzed.

8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes

By simply improving the first version of hybrid stress element method proposed by Pian, several 8- and 12-node plane quadrilateral elements, which are immune to severely distorted mesh containing elements with concave shapes, are successfully developed. Firstly, instead of the stresses, the stress function ϕ is regarded as the functional variable and introduced into the complementary energy functional. Then, the fundamental analytical solutions (in global Cartesian coordinates) of ϕ are taken as the trial functions for 2D finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. Thus, the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied. Thirdly, by using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element boundaries, which can be interpolated directly by the element nodal displacements. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid stress-function (HS-F) element is obtained. This technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy. And the first hybrid stress element proposed by Pian is just a special case within this frame. Following above procedure, two 8-node and two 12-node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of Airy stress function. Numerical results show that, the 8-node and 12-node models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle. Furthermore, these elements do not possess any spurious zero energy mode and rotational frame dependence.

Analysis of internal stresses induced by strain recovery in a single SMA fiber–matrix composite

Due to the shape memory effect (SME) of shape memory alloy (SMA) fiber and the different thermal expansion coefficients between the constituent materials, internal stresses are induced when a single prestrained SMA fiber–matrix composite is heated. To avoid interfacial debonding between the SMA fiber and the epoxy matrix and to take full advantage of the properties of SMAs, it is necessary to investigate the internal stress distribution profiles. In the present study, stress functions which satisfy the equilibrium equations in SMA fiber and matrix are employed to analyze the stress distributions in SMA composite. The principle of minimum complementary energy is utilized to solve the stress functions and the internal stress distributions are obtained. The parameter effect is also investigated and some useful results are obtained. The finite element simulation illustrates the validity of the theoretical results. The results indicate that the maximum interfacial shear stress is located at both embedded ends of the composite and the maximum axial stress on the fiber appears at the midpoint of the entire embedded length.

Stress distributions in single shape memory alloy fiber composites

Shape memory alloy (SMA) in the form of wires or short fibers can be embedded into host materials to form SMA composites that can satisfy a wide variety of engineering requirements. The recovery action of SMA inclusions induced by elevated temperature can change the modal properties and hence the mechanical responses of entire composite structures. Due to the weak interface strength between the SMA wire and the matrix, interface debonding often occurs when the SMA composites act through an external force or through actuation temperature or combination of the two. Thus the function of SMAs inside the matrix cannot be fully utilized. To improve the properties and hence the functionality of SMA composites it is therefore very important to understand the stress transfers between SMA fibers and matrix and the distributions of internal stresses in the SMA composite. In this paper, a theoretical model incorporating Brinson’s constitutive law of SMA for the prediction of internal stresses is successfully developed for SMA composites, based on the principle of minimum complementary energy. A typical two-cylinder model consisting of a single SMA fiber surrounded by epoxy matrix is employed to analyze the stress distributions in the SMA fiber, the matrix, and at the interface, with important contributions of the thermo-mechanical effect and the shape memory effect. Assumed stress functions that satisfy equilibrium equations in the fiber and matrix respectively are utilized, as well as the principle of minimum complementary energy, to analyze the internal stress distributions during fiber pull-out and the thermal loading process. The entire range of axisymmetric states of stresses in the SMA fiber and matrix are developed. The results indicate substantial variation in stress distribution profiles for different activation and loading scenarios.

A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions

A simple but robust 4-node hybrid stress-function (HS-F) membrane element with drilling degrees of freedom is developed based on the principle of minimum complementary energy. Its stress fields are derived from the first seven fundamental analytical solutions of the Airy stress function. The assumed displacements along element boundaries employ compatible mode of Allman for which the nodal drilling degrees of freedom are considered. Numerical results show that the proposed new element, denoted as HSF-Q4@q-7@b, exhibits much improved numerical accuracy and robust performance. In particular, the element performs well even when the element shape degenerates into a triangle or concave quadrangle.

Analytical trial function method for development of new 8‐node plane element based on the variational principle containing Airy stress function

Purpose − The purpose of this paper is to propose a novel and simple strategy for construction of hybrid‐“stress function” plane element. Design/methodology/approach − First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8‐node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. Findings − A new hybrid element stiffness matrix is obtained. The resulting 8‐node plane element, denoted as analytical trial function (ATF‐Q8), possesses excellent...

A novel hybrid stress-function finite element method immune to severe mesh distortion

This paper introduces a hybrid stress-function finite element method proposed recently for developing 2D finite element models immune to element shapes. Deferent from the first version of the hybrid-stress element constructed by Pian, the stress function ϕ of 2D elastic or fracture problem is regarded as the functional variable of the complementary energy functional. Then, the basic analytical solutions of ϕ are taken as the trial functions for finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. By using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element edges. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid-function element can be obtained. As examples, two (8- and 12-node) quadrilateral plane elements and an arbitrary polygonal crack element are constructed by employing different basic analytical solutions of different stress functions. Numerical results show that, the 8- and 12-node plane models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle; and the crack element can also predict accurate results with very low computational cost in analysis of stress-singularity problems.

Variational analysis of cracked angle-ply laminates

Stiffness reduction of angle-ply laminates of the type [ θ m ( 2 ) / θ n ( 1 ) ] s containing intralaminar cracks in the middle laminae is analyzed. Using the principle of minimum complementary energy an optimal admissible stress field is derived that satisfies equilibrium, boundary and traction continuity conditions. The method provides a lower bound for the stiffness matrix of a cracked angle-ply. Results for longitudinal stiffness of a cracked Glass/Epoxy [ 0 ° / 45 ° ] s laminate are in very good agreement with experimental data. Asymptotic behavior of the laminate stiffness when the crack densities are low or high is derived and its importance is discussed.

Stress analysis of laminates of carbon fiber reinforced plastics, containing transverse cracks, considering free-edge effect and residual thermal stress

A variational approach is presented to evaluate the 3-dimensional stress state in cross-ply laminates of the form [(S)/90 n ] s that contain transverse cracks in the 90° plies, where (S) is any orthotropic sublaminate. Admissible stress states that satisfy equilibrium and all boundary and interface conditions are constructed, and the principle of minimum complementary energy is employed to find an optimal approximation of the composite strain energy. Using this method of analysis, we can express the stress state by considering the free-edge effect, which causes edge delamination in cracked laminates. The calculated results using the proposed model showed good agreement with the results calculated by the finite element method.

A new method of estimating the equivalent elastic modulus of RCCD

Based on the composite material mechanic theory, the analysis method for the equivalent elastic moduli which are parallel and vertical to the construction interface of RCCD is studied in this paper. The differences between the equivalent elastic moduli which are vertical to the construction interface of RCCD gotten from different methods are discussed in detail. The variation range of the equivalent elastic modulus which is vertical to the construction interface of RCCD is studied based on the principle of minimum complementary energy and the principle of minimum potential energy. The effect of the related influential factors on the equivalent elastic modulus is analyzed. The estimation formula of the equivalent modulus which is vertical to the construction interface of RCCD is proposed. The feasibility of the approach proposed in this paper is analyzed through an example.

A Mechanical Model of 3D Braided Composites with Internal Transverse Crack

A laminated mechanical model has been developed that can predict the mechanical properties of 3D braided composites with internal transverse crack. By using the principle of minimum complementary energy, the stress components are obtained in closed-form expressions in terms of a perturbation function, which is governed by two fourth-order inhomogeneous ordinary differential equations. The effective Young’s modulus and Poisson’s ratio of 3D braided composites with internal crack are calculated using the homogenization theorem. To demonstrate the solution, some examples are analyzed. Good agreement is obtained between the presented results and the values predicted by the modified finite element method.

An Analytical Method for Interlaminar Stresses Due to Global Effects of Ply Drop-Offs

An analytical method to determine the interlaminar stresses in laminates with ply drop-offs is presented. The mechanisms that cause interlaminar stresses are identified and divided into effects due to global and local discontinuities. The former type of discontinuity arises due to the overall laminate construction, such as the differences in laminate properties from the undropped region to the dropped region and due to the termination of the plies. The latter type of discontinuity arises due to the geometric and related details of the ply drop-offs in the drop-off region. In the current work, the interlaminar stresses due to the global effects of ply drop-offs are obtained via consideration of a simplified model of a laminate with ply drop-offs based on the assumption that the drop-off zone has no effective length and that the load transfer from the terminated plies to the continuous plies occurs completely in the undropped region. Mathematically, closed-form expressions for the interlaminar stresses are obtained by assuming a priori a set of admissible stress fields that depends on known functions in the thickness direction and arbitrary functions in the lengthwise direction. The principle of minimum complementary energy along with appropriate boundary conditions are then used to obtain the final form of the interlaminar stresses. The results from the current work are validated through comparisons with results from previous finite element analysis from the literature for a free-edge problem, which the current method can also be used to analyze, and for laminates with external and internal ply drop-offs. The current method allows one to obtain the overall stress field efficiently and, thus, can be used to examine many different laminate configurations. The contribution to the overall stress field due to the global effects of ply drop-offs and its importance as well as possible extensions to obtain more accurate local stress fields are discussed.

Ana posteriorierror estimator for thep- andhp-versions of the finite element method

An a posteriori error estimator is proposed in this paper for the p- and hp-versions of the finite element method in two-dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42:561–587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non-uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p- and hp-adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd.

A mechanical model of 3D braided composites with transverse and longitudinal cracks

A simplified model of 3D braided composites with transverse and longitudinal cracks is presented. By using the principle of minimum complementary energy, the stress components are obtained in closed-form expressions in terms of a perturbation function, which is governed by two fourth-order inhomogeneous ordinary differential equations. The effective Young's modulus and Poisson ratio of 3D braided composites with transverse and longitudinal cracks are calculated using the homogenization theorem. To demonstrate the solution, some examples are analyzed.

A damage model for 3D braided composites with transverse cracking

A simplified model of three-dimensional (3D) braided composites with the transverse cracking is presented. By using the principle of minimum complementary energy, the stress components are obtained in closed-form expressions in terms of a perturbation function, which is governed by two fourth-order inhomogeneous ordinary differential equations. All possible expressions of this perturbation function are obtained in closed forms. The effective Young’s modulus and Poisson ratio of 3D damaged braided composites are calculated using the determined minimum complementary energy. To demonstrate the solution, some examples are analyzed.

Damaged mosaic laminate model of woven fabric composites with transverse yarn cracking and interface debonding

A variational solution based on the principle of minimum complementary energy is presented for a damaged mosaic laminate model of woven fabric composites. Two damage modes, i.e. transverse yarn cracking and interface debonding, are considered. The stress components in the debonded segments of the laminate are obtained explicitly, while those in the perfectly bonded laminate segments are presented in terms of a perturbation function, which is governed by two fourth-order inhomogeneous ordinary differential equations. All possible expressions of this perturbation function are obtained in closed forms. The effective Young modulus and Poisson ratio of the damaged laminate are calculated using the determined stress field and the associated minimum complementary energy. Being applicable to the woven laminate in either the plane strain or the plane stress state, with the yarn materials either orthotropic or transversely isotropic, the current closed-form solution is very suitable for parametric studies. To demonstrate the application of the solution, a parametric study of some 176 sample cases is conducted using two different composite systems (i.e. glass fiber/epoxy and ceramic fiber/ceramic), four transverse yarn crack densities and eleven debonded lengths. The numerical results obtained here for the cases involving only the first damage mode (i.e. transverse yarn cracking) are identical with those reported in an earlier paper by Gao and Mall [Int J Solids Struct 38 (2001) 855]. When both the first and second damage modes are present, the sample calculations of this study quantitatively illustrate the effects of the two co-existing types of damage on the reduction in Young's modulus and Poisson's ratio of the damaged woven laminate.

Interfacial stresses in plated beams with cracks

In this paper, an analytical interfacial stress analysis is presented for cracked reinforced concrete beams bonded with thin plates and subjected to symmetrical tensile forces and/or end bending moments. A representative element between two contiguous cracks is considered under the assumption of plane stress. By means of the principle of minimum complementary energy, the analysis provides a closed-form solution that can be used easily to predict the stress distributions near the cracks in the beams. Numerical results from the present analysis are presented to study the effects of both material and geometric properties of the beams on the distributions of the interfacial stresses.

The “free-corner effect” in thermally loaded laminates

The occurrence of localized high interlaminar stresses is a well-known phenomenon along the free edges of composite laminates, which is called “free-edge effect”. Beyond the generally considered situation of a straight free laminate edge, a more specific but also very important situation is that of a free rectangular laminate corner. Here the localized occurrence of interlaminar stresses can be named as “free-corner effect”. For this free-corner effect an approximate closed-form description is given for a class of relatively simple laminates subjected to thermal loading. The applied approach makes use of an equilibrium stress distribution and the principle of minimum complementary energy to describe the situation in the laminate corner region. The predicted interlaminar stress distribution at the ply interfaces is shown to be in good agreement with comparative 3-dimensional finite element analyses.

Stress analysis of double-strap bonded joints using a variational method

The design of bonded composite repairs to cracked structure requires not only accurate stress analysis but also the confidence that the design formulae used are truly conservative. This paper describes a stress analysis of the double-strap joint based on a modification of Hashin's variational method for orthogonally cracked composite laminates in which the principle of minimum complementary energy has been employed. The modification is essentially to introduce an adhesive layer and to include isotropic as well as orthotropic material as is typically found in bonded repairs. A simple analytical expression is found for the equivalent spring constant for the joint, which is a parameter that is useful in calculating the stress intensity factor for a patched crack. The expression is shown to yield a truly conservative estimate, in this case a lower bound, for the equivalent spring constant.

Variational solution for a cracked mosaic model of woven fabric composites

A variational solution for a cracked mosaic laminate model of woven fabric composites is presented using the principle of minimum complementary energy. The solution is derived for the woven laminate in either the plane strain or the plane stress state, with the warp/fill yarn materials being either orthotropic or transversely isotropic, unlike other existing solutions in the literature of laminate elasticity. The stress components are given in closed-form expressions in terms of a perturbation function, which is governed by two (uncoupled) fourth-order inhomogeneous ordinary differential equations (i.e., Euler–Lagrange equations) when the thermal effects are included. All possible expressions of this perturbation function are obtained in closed forms. Young’s modulus of the cracked laminate is calculated using the determined minimum complementary energy. The present closed-form solution can account for different yarn materials, applied loads (crack densities), geometrical dimensions, or their combinations. To demonstrate the solution, a total of 60 sample cases are analyzed using three different composite systems (i.e., glass fiber/epoxy, graphite fiber/epoxy and ceramic fiber/ceramic) and ten different crack densities. The obtained numerical results are also compared to two existing elasticity solutions for cross-ply laminates.