Complementary Strain Energy Research Articles

A new approach to evaluate material creep properties by C-shape ring specimen with fixed constraints

Small specimen test technique is widely used to evaluate material creep properties for those components operated at the elevated temperatures. However, the existing small specimen test techniques cannot be well applied for small miniature tubular components well, such as steam pipe and gas turbine recuperator in aviation, due to the equivalent gauge length, specimen volume, stress analysis method and conversion relationship. In this paper, combing the advantages of three-point bending specimen with fixed constraints and small ring creep specimen, a new C-shape ring specimen with fixed constraints (CRSFC) was proposed. Based on beam bending model, the conversion relationship between CRSFC and uniaxial creep test data was established by using complementary strain energy and reference stress method. Using finite element method, conversion coefficients and stress evolution process were obtained. Then the feasibility and accuracy of the above conversion relationship were verified by carrying out four sets of CRSFC creep tests of 1.25Cr0.5MoSi at 550 °C. It can be seen that creep parameters obtained by CRSFC agree well with those from uniaxial creep test with the maximum error of 22.6%. Finally, the effects of the straight edge length, indenter radius, eccentric load and indenter-specimen surface friction coefficient on the load-point steady-state displacement rate were investigated by finite element method. The results show that the effect of the straight edge length is negligible. In order to minimize the effects of other parameters, the indenter radius 0.6 mm ≤ r ≤ 1.2 mm, eccentric displacements e ≤ 0.2 mm and friction coefficient f ≤ 0.8 are recommended in CRSFC creep test.

A mixed finite element based inverse approach for residual stress reconstruction

In this paper a full-field residual stress identification procedure is proposed and explored from partially measured residual elastic strain or stress within a PDE constrained optimization framework. At first, we have constructed a penalized least-square cost functional with an appropriate regularization term. We have taken data discrepancy term in energy norm while the regularization term is a complementary strain energy functional. We then construct an appropriate Lagrangian to impose the optimization constraint. It turns out that using first order optimality condition leads to standard two field mixed, of Hellinger-Reissner type, variational formulation with residual stress and Lagrangian variable as independent field variables. This allows us to use standard two and three dimensional mixed finite elements for approximation of the stress and Lagrangian variable. Using mixed finite element set-up, we then solved these equations to obtain full-field residual stress distribution. One key issue of the proposed formulation, as in case of any inverse problem, is the choice of penalization parameter. In this context, we have used modified L-curve based approach to choose optimal penalty parameter. Finally, we assess the numerical effectiveness of the proposed procedure against a few problems of engineering interest.

Fracture analysis of a nonhomogeneous beam with two concentric cylindrical longitudinal cracks

This paper is concerned with the analysis of the fracture behaviour of a nohomogeneous cantilever beam with two concentric longitudinal cracks. The beam has a circular cross-section with linearly varying radius along the beam length. Moreover, the beam exhibits continuously varying material inhomogeneity in the radial direction. The fracture is analyzed in terms of strain energy release rate assuming nonlinear mechanical behaviour of the material. For this purpose, solutions for the strain energy release rate are derived by considering the energy balance. Two cantilever beam configurations with different lengths of longitudinal cracks are analysed. Moreover, the two cracks are arranged arbitrarily in the radial direction. The longitudinal fracture behaviour of the beam is also analysed by considering the complementary strain energy for verification. The strain energy release rate solutions are used to investigate the influence of varying radius of the cross section along the length of the beam on the longitudinal fracture behaviour. The effects of crack lengths and the location of the two concentric cracks in the radial direction on fracture are also studied. The influences of the loading conditions of the beam and the inhomogeneity of the material in the radial direction on the fracture behaviour are also evaluated.

Solutions of the interfacial shear and normal stresses in plate flexural-strengthened beams based on different complementary strain energy assumptions

A series of solutions based on five different complementary strain energy assumptions (named as Assumptions A, B, C, D, E) are developed for the prediction of interfacial shear and normal stresses in plate flexural–strengthened beams. Basic differences between the Assumptions are that in the formulating of the energy, Assumption A includes the transverse normal and shear stress fields in the beam and in the plate, Assumption B includes the normal stresses but exclude the shear stresses, Assumption C includes the shear stresses but excludes the normal stresses, Assumptions D excludes the normal stresses and the shear stresses in the beam but include the shear stresses in the plate, and Assumption E excludes the normal and shear stresses in the beam and in the plate. The energy expressions of the five Assumptions are developed based on derived statically admissible stress fields and an a generalized 2D-Hooke’s law constitutive model. Through an energy variation principle, compatibility equations and boundary conditions of the interfacial stresses are obtained. Closed form and numerical solutions are then developed for the compatibility equations. By comparing against the interfacial stresses of 2D- and 3D- finite element analyses (FEA), it is observed that Assumptions A and B significantly underpredict the interfacial stresses, Assumption C excellently predicts the interfacial normal stresses, Assumptions D and E excellently predict both the interfacial shear and normal stresses, and Assumptions (C, D, E) excellently capture the FEA stress fields (those are strongly varied across the adhesive thickness). The present study is applicable to plate-strengthened beams with general force boundary conditions but it is limited to linearly elastic orthotropic/isotropic materials.

Theoretical analysis of fracture in non-linear elastic functionally graded beam of linearly changing thickness with three longitudinal cracks

The present paper is focussed on fracture analysis of a functionally graded nonlinear elastic cantilever beam structure with three longitudinal cracks. The cross-section of the beam is a rectangle. The beam height increases linearly towards the clamped end. The material is functionally graded along the thickness of the beam. The J-integral approach is applied in the fracture analysis. The solutions to the J-integral for the three cracks are verified by deriving of the strain energy release rate. For this purpose, the complementary strain energy cumulated in the beam is differentiated with respect to the areas of the three cracks. A parametric investigation of the longitudinal fracture behaviour of the beam is carried-out by applying the solutions to the J-integral. The effects of the beam geometry, the crack length and the material gradient along the beam thickness on the longitudinal fracture behaviour are evaluated and discussed.

Extension of Castigliano\u2019s method for isotropic beams

In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

A two-level nonlinear beam analysis method

The nonlinear response of straight, constant cross-section beams is investigated by means of a two-level finite element solution procedure. The higher level model is built starting from the Hellinger–Reissner principle, with the normal stress resultant and moment resultant as additional unknowns field beside the beam reference line position and orientation field. A lower-level nonlinear cross-section problem is defined for each integration point. The two-level models are linked together by the normal stress resultant and moment resultant, in one direction, and by the variation of the complementary strain energy in the other: the cross-section level do deform in such a way that the normal stress resultant and moment resultant are equal to those of the beam model, while the higher level beam model receives the gradient and Hessian of the complementary strain energy with respect to the resultants. The complementary strain energy gradient and Hessian are computed by defining suitable first and second order adjoint problems.

Investigation of two parallel lengthwise cracks in an inhomogeneous beam of varying thickness

Analytical investigation of the fracture of inhomogeneous beam with two parallel lengthwise cracks is performed. The thickness of the beam varies continuously along the beam length. The beam is loaded in three-point bending. Two beam configurations with different lengths of the cracks are analyzed. The two cracks are located arbitrary along the thickness of the beam. Solutions to the strain energy release rate are derived assuming that the material has non-linear elastic mechanical behavior. Besides, the beam exhibits continuous material inhomogeneity along its thickness. The balance of the energy is analyzed in order to derive the strain energy release rate. Verifications of the solutions are carried-out by considering the complementary strain energy stored in the beam configurations. The influence of the continuous variation of the thickness along the beam length on the lengthwise fracture behavior is investigated. The dependence of the lengthwise fracture on the lengths of the two parallel cracks is also studied.

An accurate solution for stress fields strongly varied across the adhesive thickness in soffit plate-strengthened beams

A closed form solution and a finite difference formulation are developed for an accurate prediction of adhesive shear and normal stresses in soffit plate flexural-strengthened beams. First, three statically admissible stress fields (i.e., longitudinal normal, transverse shear and normal stresses) are derived and expressed in terms of four unknown interfacial shear and normal stresses. Then, a modified complementary strain energy is written in which the energy contributions of the transverse normal stress fields in the beam and in the plate, and the transverse shear stresses in the beam are omitted. Based on an energy variational principle, compatibility equations and corresponding boundary conditions are obtained. A closed form solution and a finite difference formulation are finally developed. Through validations, the adhesive shear and normal stresses near the plate ends predicted by the present study are in excellently agreements with those provided in several experimental and numerical studies. Also, the present study excellently captures stress fields strongly varied across the adhesive thickness, especially the transverse normal stresses. Based on the present theory, a parametric study is conducted to quantify the effects of the plate longitudinal modulus and the thickness on the interfacial shear and normal stresses. The parametric study indicates that the peak normal stresses are significantly higher than the peak shear stresses in the adhesive layer. The present study is applicable to predict the adhesive shear and normal stresses highly concentrated near the plate ends in plate-strengthened beams with general applied loads and force boundary conditions.

Longitudinal fracture analysis of inhomogeneous beams with continuously varying sizes of the cross-section along the beam length

Analyses of longitudinal fracture behavior of inhomogeneous beams which have continuously varying sizes of the cross-section along the beam length are carried-out. Beams of a rectangular cross-section are studied. It is assumed that beams exhibit continuous (smooth) material inhomogeneity along the width and height of the cross-section. A longitudinal crack located arbitrary along the beam height is analyzed. First, a cantilever beam with linearly varying width and height along the beam length is considered. The material of the beam has non-linear elastic mechanical behavior. The external loading consists of one bending moment applied at the free end of the lower crack arm. The fracture behavior is analyzed in terms of the strain energy release rate assuming that the modulus of elasticity is distributed continuously in the beam cross-section. The balance of the energy is considered in order to derive the strain energy release rate. A solution to the strain energy release rate is obtained also by considering the complementary strain energy for verification. The longitudinal fracture behavior of the inhomogeneous non-linear elastic cantilever beam configuration is studied also for the cases when the variation of the width and height of the cross-section is described by sine and power laws.

A complementary energy approach accommodates scale differences in soft tissues

The mechanics of biological entities, from single molecules to the whole organ, has been extensively analyzed during the last decades. At the smaller scales, statistical mechanics has fostered successful physical models of proteins and molecules, which have been later incorporated within constitutive models of rubber-like materials and biological tissues. At the macroscopic scale, the additive decomposition of energy functions i.e., a parallel arrangement of the tissue constituent, has been recurrently used to account for the internal heterogeneity of soft biological materials. However, it has not yet been possible to unite the mechanics at the tissue level with the actual response of the tissue components. Here, we exemplify our approach using cardiovascular tissue where the mechanical response at the tissue scale is in the range of kPa whereas the elastic modulus of collagen, the main component of the vascular tissue, is in the range of MPa GPa. In this work we develop a novel theoretical framework based on a complementary strain energy function that builds-up on a full network model. The complementary strain energy function introduces naturally an additive decomposition of the deformation gradient for the tissue constituents, i.e an arrangement in series of the constituents. We demonstrate that the macroscopic response of the tissue can be reproduced by just introducing the underlying mechanical and structural features of the micro-constituents, improving in a fundamental manner previous attempts in the mechanical characterization of soft biological tissues. The proposed theoretical framework unveils a new direction in the mechanical modeling of soft tissues and biological networks.

A Comprehensive Mathematical Study on Elastic Strain Energy and Complementary Strain Energy

A Comprehensive Mathematical Study on Elastic Strain Energy and Complementary Strain Energy

Longitudinal fracture analysis of inhomogeneous non-linear elastic rod with internal crack in torsion

The present paper deals with longitudinal fracture analysis of an inhomogeneous non-linear elastic rod with an internal longitudinal circular cylindrical crack. The rod is loaded in torsion. The cross-section of the rod is a circle. The rod exhibits continuous material inhomogeneity in radial direction. The complementary strain energy is considered in order to derive the strain energy release rate. Since the crack is internal, the torsion moments in the crack arms which are needed to obtain the complementary strain energy can not be determined explicitly. Therefore, the rod is treated as a statically undermined structure with one internal redundant (the torsion moment in the external crack arm is taken as a redundant). The solution to the strain energy release rate, derived in the present paper, is verified by considering the balance of the energy. The longitudinal fracture behavior of the rod with internal crack is compared with that of a rod with circular notch in the external crack arm. It is found that the strain energy release rate in the rod without notch is lower than that in the rod with notch in the external crack arm.

Inhomogeneous Multilayered Beams of Linearly Changing Width: a Delamination Analysis

Delamination behaviour of an inhomogeneous multilayered cantilever beam of linearly changing width of the cross-section along the beam length is analyzed. The material in each layer of the beam has non-linear elastic mechanical behaviour. Besides, each layer exhibits continuous material inhomogeneity along the layer height. The solution to the strain energy release rate derived by considering the complementary strain energy is valid for beams made of arbitrary number of layers of individual thickness and material properties. In order to verify the solution, the strain energy release rate is obtained also by considering the balance of the energy.

A stress-function variational approach toward CFRP -concrete interfacial stresses in bonded joints

This paper presents an innovative stress-function variational approach in formulating the interfacial shear and normal stresses in an externally bonded concrete joint using carbon fiber-reinforced plastic (CFRP) plies. The joint is subjected to surface traction loadings applied at both ends of the concrete substrate layer. By introducing two interfacial shear and normal stress functions on the CFRP-concrete interface, based on Euler-Bernoulli beam idea and static stress equations of equilibrium, the entire stress fields of the joint were determined. The complementary strain energy was minimized in order to solve the governing equation of the joint. This yields an ordinary differential equation from which the interfacial normal and shear stresses were proposed explicitly, satisfying all the multiple traction boundary conditions. Lamination theory for composite materials was also employed to obtain the interfacial stresses. The proposed approach was validated by the analytic models in the literature as well as through a comprehensive computational code generated by the authors. Furthermore, a numerical verification was carried out via the finite element software ABAQUS. In the end, a scaling analysis was conducted to analyze the interfacial stress field dependence of the joint upon effective issues using the devised code.

Inhomogeneous beams of circular cross-section under axial forces: a longitudinal fracture analysis

This paper deals with longitudinal fracture analysis of inhomogeneous non-linear elastic beams with continuously varying radius of the cross-section along the beam length. The external loading consists of axial forces. Beams with a longitudinal crack presenting a circular cylindrical surface are considered. The beams exhibit continuous material inhomogeneity in radial direction. A general approach for analyzing the strain energy release rate is developed by considering the energy balance. The approach is used to analyze the longitudinal fracture of an inhomogeneous non-linear elastic cantilever beam configuration. The strain energy release rate in the cantilever beam is derived also by considering the complementary strain energy for verification. The influence of the continuously varying radius of the cross-section along the length of the cantilever beam on the strain energy release rate is evaluated. The effects of the material inhomogeneity, location of the crack in radial direction, crack length and the material non-linearity on the strain energy release rate are also evaluated.

Effects of aging on lengthwise fracture in continuously inhomogeneous elastic-plastic beams: an analytical study

Effects of aging on lengthwise fracture in a continuously inhomogeneous double cantilever beam configuration which exhibits elastic-plastic behavior is studied analytically in terms of the strain energy release rate. The material is continuously inhomogeneous along the height, width and length of the beam structure. The external loading consists of two bending moments applied at the free ends of crack arms. A solution to the strain energy release rate is derived by analyzing the beam complementary strain energy with considering the aging of the inhomogeneous material. In order to verify the non-linear time-dependent solution derived, the strain energy release rate is obtained also by analyzing the balance of the energy. The solution derived is applied to investigate the variation of the strain energy release rate with the time due to the aging of the material. The effects of inhomogeneity along the height, width and length of the beam on the fracture are investigated too.

Inhomogeneous Rods in Torsion: Longitudinal Fracture Analysis by Considering the Energy Balance

Fracture of inhomogeneous non-linear elastic rods with a longitudinal circular cylindrical crack is analyzed. Rods have circular cross-section and are loaded in torsion. General approach for analyzing the strain energy release rate is developed by considering the balance of the energy. The rods exhibit continuous material inhomogeneity in both radial and length directions. The general approach is applied to analyze the strain energy release rate for a longitudinal crack in a clamped inhomogeneous rod loaded in torsion. The strain energy release rate is derived also by considering the complementary strain energy for verification.

Stationary complementary energy high-order theory for the static analysis of beams

A high-order theory is developed for the analysis of beams with general mono-symmetric cross-sections. The theory represents the nonlinear distribution of the longitudinal normal stress across the section depth by a polynomial series expansion up to any order as specified by the analyst. The corresponding shear and transverse normal stresses are obtained by satisfying the 2D infinitesimal stress-flow equilibrium conditions. The resulting statically admissible stress fields are then used in conjunction with the principle of stationary complementary strain energy to formulate the governing compatibility equations and boundary conditions. Closed form solutions are then developed for general loading and boundary conditions. Comparisons with the theory of elasticity and 3D finite element analysis predictions showcase the ability of the present theory to naturally capture shear deformation effects, transverse normal stress effects, nonlinear longitudinal normal stress distributions in deep beams, as well as the effect of support height. Unlike conventional beam solutions that are based on postulated kinematic assumptions, which tend to converge to the displacement response from below, the present theory avoids introducing any kinematic assumptions and is shown to converge to the solution from above. The theory is applicable to beams with doubly symmetric or mono-symmetric cross-sections, with isotropic or orthotropic materials, and subjected to general loading and boundary conditions. The theory is shown to offer advantages compared to other theories when modelling deep beams and/or beams with supports that are offset from the centroidal axis.

Characterizing the constitutive response of plain-woven fibre reinforced aerogel matrix composites using digital image correlation

The paper presents an experimental investigation of the constitutive behavior for plain-woven fibre reinforced silica aerogel matrix composites under a complex in-plane stress state using digital image correlation technique. Specimens were subjected to in-plane on-axis, off-axis tension and shear loadings. The material exhibits significantly nonlinear stress-strain behaviour under off-axis tension and shear loadings. Deformation mechanism of the composite material has been proposed using FEM analysis of representative volume cell qualitatively. It is proposed that nonlinear couplings of the stress and strain components may result from a combination of stress-free surface, heterogeneity and interaction of the woven fibre bundles and aerogel matrix for thin specimen under a complex stress state, even though no damage initiates in the material. The experimental results were subsequently used to develop a constitutive model based on general formulation of complementary strain energy density. The material constitutive parameters were extracted from calculated stress and measured strain using multi-variable linear least square regression. The performance of the constitutive model has been demonstrated by comparison with the off-axis tensile experimental results.