Complementary Strain Energy Research Articles

A murine experimental model for the mechanical behaviour of viable right‐ventricular myocardium

Although right-ventricular function is an important determinant of cardio-pulmonary performance in health and disease, right ventricular myocardium mechanical behaviour has received relatively little attention. We present a novel experimental method for quantifying the mechanical behaviour of transmurally intact, viable right-ventricular myocardium. Seven murine right ventricular free wall (RVFW) specimens were isolated and biaxial mechanical behaviour measured, along with quantification of the local transmural myofibre and collagen fibre architecture. We developed a complementary strain energy function based method to capture the average biomechanical response. Overall, murine RVFW revealed distinct mechanical anisotropy. The preferential alignment of the myofibres and collagen fibres to the apex-to-outflow-tract direction was consistent with this also being the mechanically stiffer axis. We also observed that the myofibre and collagen fibre orientations were remarkably uniform throughout the entire RVFW thickness. Thus, our findings indicate a close correspondence between the tissue microstructure and biomechanical behaviour of the RVFW myocardium, and are a first step towards elucidating the structure–function of non-contracted murine RVFW myocardium in health and disease.

Acoustic Resonance of a Two-Dimensional Isotropic Medium Studied Using Airy Stress Function

We have developed a theory that determines a complete set of stress field, σij, in a freely vibrating two-dimensional isotropic medium within the framework of the calculus of variation. Our formulation is based on the Airy stress function φ and the minimization of the complementary strain energy under the constrain condition || φ|| 2L2=const. By the Ritz method, the constrained variational problem becomes a linear eigenvalue problem. Numerical analysis yields 36 types of the stress functions φi. Unlike the stress fields determined from the conventional resonant ultrasound spectroscopy theory, the stress fields derived from the stress functions φi explicitly satisfy the stress-free natural boundary condition and the equilibrium equation. It is also confirmed that the 36 resonant modes can be classified into four groups according to the parity of the coefficient of the basis function. Furthermore, the stress functions φi are orthogonal in the sense of the L2 inner product. These features are similar to those of the conventional resonant ultrasound spectroscopy (RUS) theory.

Stress-function variational method for stress analysis of bonded joints under mechanical and thermal loads

High interfacial stresses near the ends of adherends are responsible for debonding failure of bonded joints used extensively in structural engineering and microelectronics packaging. This paper proposes a stress-function variational method for determination of the interfacial stresses in a single-sided strap joint subjected to mechanical and thermal loads. During the process, two interfacial shear and normal (peeling) stress functions are introduced, and the planar stresses of adherends of the joints are expressed in terms of the stress functions according to the static equilibrium equations. Two coupled governing ordinary differential equations (ODEs) of the stress functions are obtained through minimizing the complementary strain energy of the joints and solved explicitly in terms of eigenfunctions. The stress field of the joints based on this method can satisfy all the traction boundary conditions (BCs), especially the shear-free condition near the adherend ends. Compared to results based on finite element method (FEM) and other analytic methods in the literature, the present variational method is capable of predicting highly accurate interfacial stresses. Dependencies of the interfacial stresses upon the adherend geometries, moduli and temperature are examined. Results gained in this study are applicable to scaling analysis of joint strength and examination of solutions given by other methods. The present formalism can be extended conveniently to mechanical and thermomechanical stress analysis of other bonded structures such as adhesively bonded joints, composite joints, and recently developed flexible electronics, among others.

Reliability based design of frames with limited residual strain energy capacity

The aim of this paper is to create new type of plastic limit design procedures where the influence of the limited load carrying capacity of the beam-to-column connections of elasto-plastic steel (or composite) frames under multi-parameter static loading and probabilistically given conditions are taken into consideration. In addition to the plastic limit design to control the plastic behaviour of the structure, bound on the complementary strain energy of the residual forces is also applied. If the design uncertainties (manufacturing, strength, geometrical) are taken into consideration at the computation of the complementary strain energy of the residual forces the reliability based extended plastic limit design problems can be formed. Two numerical procedures are elaborated. The formulations of the problems yield to nonlinear mathematical programming which are solved by the use of sequential quadratic algorithm.

Reliability based limit design of steel frames with limited residual strain energy capacity

AbstractThe aim of this paper is to take into consideration the influence of the limited load carrying capacity of the connections on the plastic limit state of elasto‐plastic steel (or composite) framed structures under multi‐parameter stochastic loading and probabilistically given conditions. In addition to the plastic limit design to control the plastic behaviour of the structure, bound on the complementary strain energy of the residual forces is also applied. This bound has significant effect for the load parameter. If the design uncertainties (manufacturing, strength, geometrical) are expressed by the calculation of the complementary strain energy of the residual forces a reliability based extended limit design problem is formed. The formulations of the problems yield to nonlinear mathematical programming which are solved by the use of sequential quadratic algorithm. The bi‐level optimization procedure governed by the reliability index calculation. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A novel, high-sensitivity, small specimen creep test

This paper describes a novel, high-sensitivity, ring-type of small specimen creep test method, which can be used to obtain accurate creep strain data. A full theoretical description of the test technique is given; this is based on the complementary strain energy approach, which leads to an analytical solution for the load line deformation of an elliptical ring. Using the analytical solution, a reference stress approach is used to establish the conversion relationships between the applied load and the equivalent uniaxial stress and between the experimentally measured creep deformations and the equivalent uniaxial creep strains. The main features of the test method are: (a) the ring specimen has a significantly larger equivalent gauge length (EGL), when compared with that of other small specimen types; (b) the method is suitable for testing at lower stresses compared with commonly used current small-specimen test methods (this is because relatively low strains can be obtained from relatively large deformations); (c) specimens have simple geometries and the tests are easy to perform, and (d) the conversion relationships are material independent and practically insensitive to geometry change due to deformation. Experimental validation of the test method is made using the results obtained from creep tests for a P91 steel at 650 °C. Practical applications and future exploitation of the technique are addressed.

Reliability based limit analysis of steel frames with limited residual strain energy capacity

AbstractThe main structural elements of steel framed multi–storey structures are the columns, the beams and their connectionsmagenta, which can be considered rigid, pinned or semi–rigid. The aim of this paper is to analyze the effect of the semi–rigid connections on the limit analysis of steel structures under multi–parameter static loading where to control the plastic behavior of the structure a bound on the complementary strain energy of the residual forces is applied. The bound value is assumed to be uncertain and is modelled by a random variable. By the use of this constraint a reliability based problem formulation is constructed. The formulation of the problem yields a nonlinear mathematical programming with a nested algorithm. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Influence of strain rate on fracture behavior of poly(methyl methacrylate)

The effect of strain rate on fracture behavior of poly (methyl methacrylate) was investigated. The uniaxial tensile rupture tests for the poly (methyl methacrylate) samples were carried out at different strain rates at ambient temperature. It is found that the elastic modulus of the material increases with increasing strain rate, while the elongation is reversal with strain rate. Simultaneously, there exists a critical strain rate within which the stress-strain curves overlap one another, and beyond which the curves depart from each other. The amount of energy added to the system due to work done by the imposed load was calculated, and the strain energy stored in the material at each strain rate was calculated by the current stress integral with respect to strain. The complementary strain energy, which is the difference between the work and the strain energy, was obtained and was considered to supply the surface energy to create a new crack surface in the polymeric material. It is found that the work done by the imposed load, which is needed for the fracture of poly (methyl methacrylate) sample, decreases with increasing strain rate, and the strain energy decreases with strain rate as well, which demonstrates that the polymeric material at high strain rate is easier to fracture than that at low strain rate. As the strain rate increases, the fracture mode changes from ductile, semi-ductile to brittle mode. The complementary strain energy almost sustains a constant at any strain rate. The density of surface energy, which characterizes the energy per unit area needed for creating crack surface, is a strain rate-independent material constant.

A decomposition of the strain energy release rate associated with the initiation of transverse cracking, longitudinal cracking and delamination in cross-ply laminates

An energy criterion is proposed to study the damage evolution in a composite cross-ply laminate. This criterion is based on the computation of the partial strain energy release rate associated with each damage mechanism (transverse cracking, longitudinal cracking and delamination) and mode (I, II or III). Several decompositions of the complementary strain energy are put forward. Each component part of this decomposition is related to a specific damage mechanism and loading mode. The related criterion can predict and describe the initiation and propagation of the different damage mechanisms.

Free-edge stresses and progressive cracking in surface coatings of circular torsion bars

This paper considers the explicit solutions of free-edge stresses near circumferential cracks in surface coatings of circular torsion bars and their application in determining the progressive cracking density in the coating layers. The problem was formulated within the framework of linear elastic fracture mechanics (LEFM). The free-edge stresses near crack tip and the shear stresses in the cross-section of the torsion bar were approached in explicit forms based on the variational principle of complementary strain energy. Criterion for progressive cracking in the coating layer was established in sense of strain energy conservation, and the crack density is thereby estimated. Effects of external torque, aspect ratio, and elastic properties on the density of progressive cracking were examined numerically. The present study shows that, in the sense of inducing a given crack density, compliant coating layer with lower modulus has much higher critical torque than that of a stiffer one with the same geometries and substrate material, i.e., compliant coating layer has greater cracking tolerance. Meanwhile, the study also indicates that thicker surface coating layer is more pliant to cracking than the thinner ones. The present model can be used for analyzing the damage mechanism and cracking tolerance of surface coatings of torsion shafts and for data reduction of torsional fracture test of brittle surface coatings, etc.

The influence of semi‐rigid connections on the shakedown of elasto‐plastic frames

AbstractThe main structural elements of steel framed multi‐storey structures are the columns, the beams and their connections. The assumption has been widely applied in the past that the connections are either rigid or pinned. The actual behaviour of the connections is however somewhere between these limits, they are semi‐rigid. The aim of this paper is to analyze the effect of the semi‐rigid connections on the shakedown of steel structures under multi‐parameter static loading. In the analysis, to control the plastic behavior of the structure, bound on the complementary strain energy of the residual forces is applied. The formulation of the problem yields to nonlinear mathematical programming. The results of the numerical calculation show that the semi‐rigid connections can influence significantly the magnitude of the shakedown parameter. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Simple and efficient interlaminar stress analysis of composite laminates with internal ply-drop

A stress function-based variational method is developed to investigate the interlaminar stresses near the dropped plies. The present method is based on expanding stress functions in terms of harmonic series to the out-of-plane direction. The stresses satisfying the traction free boundary conditions of tapered laminates are obtained by the complementary strain energy principle. Stress concentrations near the dropped plies are predicted well as the number of initially assumed out-of-plane stress functions is increased. The proposed method is relatively simple and efficient in the prediction of interlaminar stresses near the dropped plies. Therefore, it is expected that the proposed method can be used in the parametric study of the preliminary design stage of tapered composite laminates.

On the stress function approach in three-dimensional elasticity

Stress function-based solution procedures in elasticity require the knowledge of those stress functions that give zero complementary strain energy. In the two-dimensional case, the structure of the zero-energy first-order stress functions is as simple as that of the zero-energy displacements, and the number of the zero-energy modes is three [1]. This paper investigates the more complicated three-dimensional case and, considering a possible set of six independent first-order stress functions, derives the zero-energy stress functions in polynomial form. It is pointed out that the number of the zero-energy stress function modes in the three-dimensional case is infinite.

A contact finite element algorithm for the multileaf spring of vehicle suspension systems

This paper presents an innovative finite element (FE) algorithm for the contact problem of the multileaf spring in vehicles. The well-established classic beam theory is adopted to construct the complementary strain energy variational. A piecewise contact stress pattern is approximated to the real contact state between two layered beams. The vector of nodal contact stresses is taken to represent primary state variables. To implement the principle of the least complementary energy, a quadratic programming (QP) problem with equality and unilateral constraints is formulated. The corresponding Kuhn-Tucker condition is equivalent to the linear complementary problem. In this study, Lemke's algorithm is applied to solve for the nodal stress vector and subsequently to determine the contact stress distribution over the contact surfaces between the layered beams. The algorithm is verified against experimental stress analysis, and it is found that the computation and test correlate well.

直接法による永久変形量の評価法について

Direct methods are to estimate a load factor and a displacement field at the ultimate limit state or shakedown limit state by the use of limit theorems, the shakedown theorems or other energy principles, instead of the evolutionary method which is a step by step elasto-plastic calculation in the time domain. A mathematical and mechanical structure of a direct method for the estimation of permanent deformation under the repeated loads based on the minimisation of the complementary elastic strain energy is investigated in this paper. A methodology is similar to the author's research works on the hybrid type limit / shakedown analysis alogorithms. From the point of Lagrangian duality theory, both static and kinematic analyses are derived from the proposed Lagrangian. Characteristics of the proposed method are also discussed

Force Method Revisited

Computational mechanics is an important field in engineering with a strong foundation in continuum mechanics and finite element analysis. The standard and integrated force methods are reviewed, and a structural analysis technique using the finite element force method based on the complementary strain energy is proposed. Similar to the standard force and the integrated force methods, the force method based on the complementary strain energy uses the same equilibrium equations. However, the compatibility conditions are satisfied through the complementary strain energy. The Gauss elimination technique has been employed successfully to generate automatically a basis determinate structure and redundant members. Extension of the force method to dynamics and benchmark problems with displacement-stress and frequency constraints are presented.

Plastic behaviour and stability constraints in the shakedown analysis and optimal design of trusses

A shakedown analysis and optimum shakedown design of elasto-plastic trusses under multi-parameter static loading are presented. To control the plastic behaviour of the truss, bounds on the complementary strain energy of the residual forces and on the residual displacements are applied and for the bars under compression critical stresses updated during the iteration are taken into consideration. The formulation of problems is suitable for nonlinear mathematical programming which is solved by the use of an iterative procedure. The application of the method is illustrated by three test examples.

LAYOUT AND SHAPE OPTIMIZATION OF ELASTOPLASTIC DISKS WITH BOUNDS ON DEFORMATION AND DISPLACEMENT1*,2†

ABSTRACT A design method is presented for layout optimization of linearly elastic–perfectly plastic disks subjected to multiparameter loading. The method is based on the static theorem of shakedown analysis and on the concept of porous material, where the material distribution of the discretized structure is described by the densities of the finite elements that are considered design variables. In addition to shakedown, two further compliance constraints are applied: bounds for the complementary strain energy of the residual stresses, and for the residual displacements. By the proper choice of these bounds the plastic behavior of the disk can be controlled and in special cases the elastic and fully plastic optimal solutions can be determined. The formulation of this problem yields to nonlinear mathematical programing. The application is illustrated by numerical examples. The method can also be applied to trusses, frames, and plates. *Communicated by M. Botkin. †Presented at ICTAM2000, held in Chicago in September 2000.

Spring constants of filament-wound composite circular rings

Closed-form solutions from complementary strain energy are derived for the spring stiffnesses of mid-surface symmetric, filament-wound, composite circular rings under unidirectional loading. A three-dimensional finite element analysis (FEA) including the effects of transverse shear has also been applied to study the problem. Four > 45° and four > 75° E-glass/epoxy composite rings of odd numbers of covers were tested. Comparisons of the results obtained from the two methods with experimental data are made and the results are found to be in good agreement. The FEA prediction of stiffness is always higher than the theoretical result. The relationships between the spring stiffnesses and the winding angles and geometry of the filament-wound composite ring are considered and discussed.

Optimal plastic limit and shake-down design of bar structures with constraints on plastic deformation

Optimal plastic design methods of discretized bar structures are presented where the objective function is the volume of the structure and the permanent deformations are controlled by introducing a constraint on the complementary strain energy of the residual internal forces. Single- and multiparameter loadings are considered and in the latter case the constraint on shakedown is also included in the optimal design procedure. Alternative formulations of the problems, where the complementary strain energy of residual forces is minimized at a given volume of the structure, are also presented. The proposed design methods are nonlinear and their solutions lead to nonlinear mathematical programming. Iterative solution techniques can also be efficiently applied. This is illustrated by an example.