Complementary Strain Energy Research Articles

Nonlinear Elastic Behavior of Unidirectional Composites With Fiber Waviness Under Compressive Loading

Nonlinear elastic behavior of unidirectional composites with fiber waviness under compressive loading was investigated theoretically and experimentally. Unidirectional carbon/epoxy composites with uniform, graded, and localized fiber waviness were studied. Complementary strain energy was used to derive the material nonlinear stress-strain relations. Nonlinear material properties obtained from shear and longitudinal and transverse compression tests were incorporated into the analysis. Compression tests of specimens with known fiber waviness were conducted to verify the constitutive relations. Experimental results were in good agreement with predictions based on the constitutive model.

Simplified Model of Unilateral Damage for RC Frames

This paper presents an approach based on the concepts of continuum damage mechanics and standard inelastic analysis of reinforced concrete (RC) frames, modeling the hysteretic behavior of RC frames. A frame member is considered as the assemblage of an elastic beam-column and two inelastic hinges, as in the traditional lumped plasticity models. Three sets of internal variables are introduced that measure plastic rotations and state of damage of the member. Proposed as functions of the internal variables are an expression for the flexibility matrices and the complementary strain energy of a member. A resulting unilateral model is verified in simulations of various experiments for which data were available in the literature. The model is very simple and consequently not comprehensive; however, the technique is amenable to further generalizations.

Universal Bounds for Overall Properties of Linear and Nonlinear Heterogeneous Solids

For a sample of a general heterogeneous nonlinearly elastic material, it is shown that, among all consistent boundary data which yield the same overall average strain (stress), the strain (stress) field produced by uniform boundary tractions (linear boundary displacements), renders the elastic strain (complementary strain) energy an absolute minimum. Similar results are obtained when the material of the composite is viscoplastic. Based on these results, universal bounds are presented for the overall elastic parameters of a general, possibly finite-sized, sample of heterogeneous materials with arbitrary microstructures, subjected to any consistent boundary data with a common prescribed average strain or stress. Statistical homogeneity and isotropy are neither required nor excluded. Based on these general results, computable bounds are developed for the overall stress and strain (strain-rate) potentials of solids of any shape and inhomogeneity, subjected to any set of consistent boundary data. The bounds can be improved by incorporating additional material and geometric data specific to the given finite heterogeneous solid. Any numerical (finite-element or boundary-element) or analytical solution method can be used to analyze any subregion under uniform boundary tractions or linear boundary displacements, and the results can be incorporated into the procedure outlined here, leading to exact bounds. These bounds are not based on the equivalent homogenized reference solid (discussed in Sections 3 and 4). They may remain finite even when cavities or rigid inclusions are present. Complementary to the above-mentioned results, for linear cases, eigenstrains and eigenstresses are used to homogenize the solid, and general exact bounds are developed. In the absence of statistical homogeneity, the only requirement is that the overall shape of the sample be either parallelepipedic (rectangular or oblique) or ellipsoidal, though the size and relative dimensions of the sample are arbitrary. Then, exact analytically computable, improvable bounds are developed for the overall moduli and compliances, without any further assumptions or approximations. Bounds for two elastic parameters are shown to be independent of the number of inhomogeneity phases, and their sizes, shapes, or distribution. These bounds are the same for both parallelepipedic and ellipsoidal overall sample geometries, as well as for the statistically homogeneous and isotropic distribution of inhomogeneities. These bounds are therefore universal. The same formalism is used to develop universal bounds for the overall non-mechanical (such as thermal, diffusional, or electrostatic) properties of heterogeneous materials.

An Expression for the Strain Energy of Laminated Composite Thin Shells

An expression for the complementary strain energy of a linear elastic laminated composite thin-walled structure is presented which describes the energy in terms of midplane properties and subsequently stress and moment resultants rather than local stress and strain values. This expression is particularly useful for the analysis of statically indeterminate structures and energy storage devices such as springs.

On the extension of Crotti's theorem to the theory of finite elastic displacements

This paper deals with the extension of the classical Crotti's theorem to the theory of finite elastic displacements. In its classical formulation Crotti's theorem is based on the complementary work of deformation of the so-called internal forces and is restricted to Eulerian treatment of the geometrically linearized theory of elasticity. In order to investigate the conditions for the extension of Crotti's theorem to the theory of finite elastic displacements, a Lagrangian description of deformation is assumed and the classical Green-Lagrange tensor is adopted as measure of strain. The internal forces are expressed in terms of the Kirchhoff-Trefftz stress tensor, which is energetically conjugate to the Green-Lagrange strain tensor. The expression corresponding to the complementary work of applied forces in terms of internal stresses differs from the integral of the complementary strain energy density by the addition of a term yielded by the product of displacement gradients which, of course, vanishes in the geometrical linearized theory. The conditions required so that this expression is a perfect differential and is therefore employable in the extension of Crotti's theorem to the finite displacement theory of elasticity are then determined. Furthermore, by a general technique of calculus of variations it is shown that the compatibility conditions of the problem are implicitly embedded in this extended formulation of the theorem, as happens in the classical one. Finally, an application to a very simple example in the theory of structures is discussed.

An investigation into the characteristics of the PG canine retraction spring

The forces and couples produced by a model of the PG maxillary canine retraction spring (Danish loop) when activated have been derived by using the complementary (strain) energy method. The theoretical results of the analysis have been verified by experiments on enlarged planar models in a special jig that, by means of strain gauges, was capable of recording the forces and couples developed when the models were activated. Good agreement between theory and experiment was obtained that enabled the analysis to be used with confidence to predict the characteristics of clinical-size loops, and to compare the behavior of the PG retractor with some other maxillary canine retraction springs.

Investigation into the behavior of symmetrically and asymmetrically activated face-bows

The distal and lateral forces and also the couples (in a transverse plane) delivered by the inner terminals of a face-bow have been derived analytically according to the complementary (strain) energy method for various face-bow configurations on the assumption that the terminals are rigidly held by the molar tubes and that the external forces acting are in the plane of the inner bow. The results of the theory have been checked experimentally with a specially designed apparatus in which two mild steel transducer pegs that carry three pairs of strain gauges were used to measure the force system delivered by the inner bow under a range of loading, both symmetrical and asymmetrical. Good agreement with the theory was obtained, confirming the validity of the theoretical analysis. It is concluded that force systems are dependent on bow geometry and that with increasing asymmetry of the external loading transference of the distal load to the power-arm side is accompanied by a shift of lateral force and couple to the opposite side.

The characteristics of the Ricketts maxillary canine retractor.

The characteristics of the Ricketts maxillary canine retraction spring (Ricketts, 1974) have been derived using the complementary (strain) energy method. The theoretical predictions of the analysis have been confirmed by experiments on large planar models in a jig fitted with strain gauges capable of measuring the forces and couples developed when the springs were activated in a way simulating clinical use. The verified analysis was used to compare the horizontal stiffness and anti-tilt couple to retraction force ratios of the Ricketts with other simpler retraction components. The behaviour of the Ricketts mandibular sectional, as predicted by the analysis was also examined.

The mechanics of asymmetric retraction loops used in fixed appliance therapy

In order to move a tooth bodily without tipping, both a force P and an anti-tilt couple N of a suitable constant magnitude are required. Although many different spring components are available to orthodontists for this purpose, a comparison of the merits of different designs has not been possible because analytical solutions to the deformation behaviour have only been generally available for the atypical case in which the loop is centrally positioned. The asymmetric case, with two redundancies, for open and closed U-loops both with or without helices has been solved using the complementary (strain) energy method on the assumption that the effect of a longitudinal tension or compression on the bending of a cantilever beam by lateral forces (the tie-bar effect) is small. Unless the arms are of equal length it is then found that unequal couples M and N act at the posterior and anterior ends of the appliance, and in addition equal but oppositely directed forces Q are induced at either end. Check measurements were carried out on enlarged model components using a special jig utilizing a mixed dead-load and strain-gauge system. Agreement with theory was generally within the known errors of measurement. The main characteristics of these retraction components in the clinical situation can consequently be predicted. The characteristics examined are the lateral and vertical stiffnesses ( S and dQ dy , respectively), the N P and Q P ratios, the self-righting ability (SRA) and dN dβ , the change in anti-tilt couple per degree change in gable angle (the angle of the mesial arm with respect of a horizontal in the unactivated appliance).

The generalized variational principles about blending energy form of non-linear elasticity

First of all, this paper gives Legendre transformation for the so-called partial corresponding variables of strain energy function Σ(Eij) and complementary strain energy function ΣC(Sij) of the elastic material, and introduces the corresponding blending complementary strain energy function Σklc and blending strain energy function Σkl. Moreover, a series of generalized variational principles of the corresponding blending energy form of non-linear elasticity is given. As a special case, there exist corresponding results[1] in linear elasticity.

Dual conservation laws in elastostatics

Several new dual conservation laws are given in this study under the assumption of the existence of the complementary strain energy density function. They are valid for both finite deformation and infinitesimal deformation of elastic solid. Applications of these laws to fracture mechanics are discussed.

OPTIMUM THERMOELASTIC DESIGN OF LAMINATED PLATES

A thermoelastic optimization procedure is proposed for the design of symmetric. laminated, fiber-reinforced plates. A maximum stiffness criterion is employed, wherein the ply orientations are found by minimizing the complementary strain energy. Approximate solutions for the thermoelastic response are obtained using the Rayleigh-Ritz procedure. Optimization is performed via a quasi-Newton method, with the initial point selected using a random jump routine. Numerical results are presented for simply supported plates subject to a temperature distribution that varies linearly in the thickness direction. Results are given for various composite materials, and the optimum thermoelastic solutions are compared with designs for mechanically loaded laminated plates.

Mechanical Performance Model for Roofing Membranes

The mechanical performance of built‐up roofing membranes, fully bonded to an underlying deck or substrate was modeled. Both linear and nonlinear stress‐strain behaviors were considered in the model development. The model is compared to previously developed models. It was found that the equality of the complementary strain energy of the fabric or felt layer with the strain energy of the bonding adhesive or bitumen layer governs both the conditions under which membrane integrity is lost and the mode of failure. Failure can occur either by membrane splitting or adhesive disbonding. The testing criteria developed are applied to a sample case.

On non-linear Beltrami-Mitchell equations

Beltrami-Mitchell equations for non-linear elasticity theory are derived using the first Piola-Kirchhoff stress and the deformation gradient tensors as field variables so as to yield linear equilibrium and compatibility equations, respectively. In the derivation it is assumed that a strain energy density and, correspondingly, a complementary strain energy density exist, and satisfy the axiom of objectivity. Substitution for the deformation gradient in the compatibility equations yields non-linear differential equations in terms of the first Piola-Kirchhoff stress tensor which may be regarded as the Beltrami-Mitchell equations of non-linear elasticity. The equations are also derived for “semi-linear” isotropic elastic materials and the theory is illustrated by three simple examples.

Hybrid-Stress Reduced Mindlin Isoparametric Elements for the Analysis of Thin Plates

ABSTRACT Hybrid-stress isoparametric finite elements are developed for the analysis of thin plates utilizing independent interpolations for transverse displacement and cross-section rotations. The formulation is based on earlier hybrid-stress plate elements that utilize Mindlin-type displacement fields and include transverse shear stress and transverse normal stress contributions. By neglecting the contributions of transverse shear and normal stresses to the complementary strain energy, isoparametric thin plate elements are obtained that have the same performance characteristics as the non-reduced element in the thin plate limit. This is illustrated for a two-node beam element. In addition to the ease of formulating isoparametric thin plate elements using C ° displacement interpolations, the primary advantage of this approach over the nonreduced element is the significant savings of computation time for element stiffness generation that is made possible. Computational considerations are illustrated for an 8-node invariant isoparametric plate element.

Vibration of Cylindrical Shells by Hybrid Finite Element Method

f tj*Kds+ f jsp Jst U) p = 1, 2, . . . n where the kinetic energy T is a function of momentum vector r£ and the complementary strain energy 17* is a function of impluse tensor T^ (t£j-.= stress tensor). The remaining terms denote the work obtained by the inner product of the boundary velocities % and surface impulses f f . In nH both TO-, within the elements, and ww, on the boundaries of the element, are subject to variation. The variations of TO must satisfy the equilibrium equations while those of ubi must conform to conditions of interelement compatibility. Equilibrium along interelement boundaries and the conditions of compatibility within the elements tend to be satisfied a posteriori through the process of extremization. In discretizing the functional nH the surface velocities are written in terms of some nodal velocities qt as {ub} = \_L-]{q} (2) The elements of matrix [L] are the appropriate interpolation functions. The interpolation functions for the impulses TOare written in two parts W = WiWi + W2{/»}2 (3) The first part [4]i{/?}i, when substituted into the equilibrium equation Ttj j = r-, will yield zeros on the right-hand side, i.e., this part furnishes the homogeneous solution of equilibrium equation. The second part, when substituted into the equilibrium equation will yield the appropriate interpolation functions for rt. The interpolation functions for tt are of course obtained through the equation tt = T^-. Finally when nH is discretized its variation can be expressed in terms of variations of /?£ and those of qj. Since the $ in one element are independent from those in the other elements, the coefficients of their variations can be equated to zero for stationary conditions of nH. The vanishing of these coefficients establishes a set of relations between /?. and qj by means of which ftt can be eliminated from nH. Finally the elements are connected and the element nodal velocities qj are

An application of the Lagrangian multiplier method to a mixed boundary value problem

This paper illustrates how use of a Lagrangian multiplier may satisfy exactly (nonholomonic) boundary conditions. By using the method presented thermoelastic investigations using complementary strain energy methods for bodies with traction free surfaces can be extended to those with mixed boundary conditions. For example, the stresses resulting from a uniform temperature change are determined in a semi-infinite slab, which is in the state of plane strain, where one lateral surface is held fixed. Previous attempts to solve the problem show that the complementary energy method does not necessarily satisfy the displacement boundary conditions.

Energy Theorems and Structural Analysis

IT is natural in reviewing the developments of Sections 3 and 4 to inquire if it is possible to enlarge upon the conception of complementary work and strain energy in a similar way as accomplished for work and strain energy by the introduction of virtual displacements.