Principle Of Minimum Strain Energy Research Articles

Modeling compliant bistable mechanisms: An energy method based on the high-order smooth curvature model

This paper applies an energy method based on the high-order smooth curvature model to address the challenges of kinetostatically modeling the complicated nonlinear post-buckling behavior of inclined compliant beams. The proposed energy method is grounded in the principle of minimum strain energy, implying that the total strain energy is minimized at the equilibrium configuration. In this work, the high-order smooth curvature model is adopted to accurately model the bending strain energy of large deformation beams. Subsequently, the Lagrange multiplier method is employed to ascertain the minimum of strain energy while concurrently determining the corresponding tip loads. Additionally, the deformation shape and maximum stress are determined via the smooth curvature model. The proposed method is introduced for the first time in modeling compliant bistable mechanisms, and it is proven that the method can be used for modeling compliant beams with inclined angles ranging from 0 to 90 degrees. Following the modeling, finite element analysis and experimental tests are conducted to verify the accuracy of the proposed energy method. A comprehensive comparative analysis between proposed method and existing methods is conducted. The comparison results prove that the model is more computationally efficient without compromising modeling accuracy The proposed modeling method can not only be used for modeling compliant bistable mechanisms but also has extendable applications in modeling initially-curved compliant beams, contact-aid design problems, and distributed load problems.The present work not only advances the practical application of the proposed energy method but also lays the foundation for future research in modeling compliant mechanisms.

Construction of strut‐and‐tie models for members under complex stress states based on topology optimization

AbstractThe construction of strut‐and‐tie models (STMs) is the key, but also the difficulty when the model is used to describe the force mechanism and guide the reinforcement layout design of concrete disturbed regions (D‐regions). Topology optimization is a promising way to guide STMs construction. Different from the existing construction methods of predetermining nodes, the angle of inclination of the truss elements is first determined to ensure that it is consistent with that of the principal stresses in a 2D finite element model of the structure. The constructed model can well reflect the force transmission path of structural members, and it is better than the existing methods when it is evaluated through the principle of minimum strain energy. To verify the validity of the STM, a reinforcement layout scheme is adopted completely according to the position of the tension ties. On this basis, another reinforcement layout scheme adds additional transverse reinforcements in the bottle‐shaped stress field. The nonlinear finite element comparative analysis of an irregular deep beam designed by the two schemes is carried out. The results show that the two ultimate loads are close. Brittle failure occurs in the former, while the latter has higher deformability and eventually ductile failure occurs.

On the uniform torsional rigidity of square concrete-filled steel tubular (CFST) sections

There are a number of scenarios in which structural members experience torsion, including under the direct application of torsional loading, when transverse loading is applied at an eccentricity to the shear centre and as a second order effect arising from lateral torsional instability. To date, the torsional rigidity of concrete-filled steel tubular (CFST) sections has yet to be fully explored; hence a study into the uniform torsional rigidity of square CFST sections is presented herein. First, the strain energy of square CFST sections is formulated, in which the longitudinal warping displacement is assumed to have an undetermined constant. The undetermined constant is then deduced by means of the principle of minimum strain energy, and thus an analytical expression for the uniform torsional rigidity of square CFST sections is obtained. The accuracy of the derived formula is verified against existing theoretical solutions for simplified scenarios, test data and the results of numerical simulations. Finally, the influence of the key parameters in the derived formula for the torsional rigidity of square CFST sections are analysed, and a simplified design formula is presented.

Development of strut‐and‐tie models using topology optimization based on modified optimal criterion

AbstractStrut‐and‐tie models (STMs) have been widely used to design disturbed regions in structural concrete members. Recently, topology optimization methods, based on the principle of minimum strain energy directly and indirectly, have been adopted to generate STMs in reinforced concrete structures. Unfortunately, current topology optimization methods do not work in many typical disturbed regions. The optimization history of the rectangular anchorage zones shows that the principle of minimum strain energy is a necessary condition for STM, not a sufficient condition. Because, it fails to capture the load transfer path properly in structural concrete tensile regions. To address this issue, a modified optimality criterion is proposed to illustrate the load path in tensile regions. And the desired STM is the one with the maximum overall stiffness and minimum strain energy of the tie. The STM of the rectangular anchorage zones and dapped beams have been developed successfully, and verified by the current ACI 318‐08 code recommendations and experimental evidence.

Kinematic modeling and solution of rigid-flexible and variable-diameter underwater continuous manipulator with load

AbstractThis paper presents a method to solve the kinematics of a rigid-flexible and variable-diameter continuous manipulator. The multi-segment underwater manipulator is driven by McKibben water hydraulic artificial muscle (WHAM). Considering the effect of elasticity and friction, we optimized the static mathematical model of WHAM. The kinematic model of the manipulator with load is established based on the hypothesis of piecewise constant curvature (PCC). We developed an optimization algorithm to calculate the length of the WHAMs according to the principle of minimum strain energy and obtain the configuration space parameters of the kinematic model. Based on the infinitesimal method, the homogeneous transformation matrices of the variable-diameter bending sections are computed, and the terminal position and attitude are obtained. In this paper, we studied the working space of the manipulator by quantitative analysis of the impact factors including pressure and load. A deep neural network (DNN) with six hidden layers is designed to solve inverse kinematics. The forward kinematic results are used to train and test the DNN, and the correlation coefficient between the output and target samples reaches 0.945. We carried out an underwater experiment and verified the effectiveness of the kinematic modeling and solution method.

ON THE OPTIMAL STRUT-AND-TIE MODELS AND DESIGN APPROACH FOR THE CABLE-PYLON ANCHORAGE ZONE

The cable-pylon anchorage zone is a typical D-region in a cable-stayed bridge, for which there has been no uniform simplified design method until now. In this paper, based on the extensive statistics of actual projects, topology optimization techniques and principle of minimum strain energy, two precise strut-and-tie models for the cable-pylon anchorage zone are proposed, which can clearly reveal the load-transmitting mechanism of the anchorage zone. Th e explicit geometric parameters of the strut-and-tie models are derived; thus, the designers can directly use these models. A simple design procedure to deploy prestressing tendons in the anchorage zone is also introduced, whose effectiveness and convenience are demonstrated by two design examples. A new design named the “one-way prestressing tendons PC cable-pylon” is also discussed regarding its application scope.

Bi-stable states of initially stressed elastic cylindrical shell structures with two piezoelectric surface layers

A theoretical model is proposed in this paper to predict the bi-stable states of initially stressed cylindrical shell structures attached by surface anisotropic piezoelectric layers. The condition for existence of bi-stability of the shell structural system is presented and analytical expressions for corresponding rolled-up radii of the stable shell are given based on the principle of minimum strain energy. The resulting solution indicates that the shell system may have two stable configurations besides its initial state under a combined action of the actuating electric field and initial stresses characterized by the bending moment. If the piezoelectric layer materials act as only sensor materials without the actuating electric field, initial stresses may produce the bi-stable states, but one corresponding to its initial state. For the shell without initial stresses, the magnitude in the actuating electric field determines the number of the stable states, one or two stable configurations besides the initial state. The theoretical prediction for the bi-stable states is verified by finite element method (FEM) simulation by using the ABAQUS code.

Optimization Research on Geometric Parameters of Scallop-Shaped Lattice Shells

<p>Free-form and bionic spatial shells are popular in the area of spatial structures. Scallop-shaped surface is the product of evolution and a kind of spatial shells that can satisfy the mechanical requirements. Based on the scallop-shaped lattice shells, this paper focused on the optimization of geometric parameters. The principle of minimum strain energy was applied to conclude the influence law of the geometric parameters on mechanical properties. Finally the optimal values of geometric parameters were obtained. The results show that the optimization of geometric parameters presents the integrated significance to improve scallop-shaped lattice shells.</p>

Optimal three-dimensional strut-and-tie models for anchorage diaphragms in externally prestressed bridges

As a typical D-region in externally prestressed bridges, an anchorage diaphragm is usually severed as the end anchorage of external tendons. In this paper, an optimal three-dimensional strut-and-tie model for the anchorage diaphragm is proposed, and direct equations to facilitate the design are also developed. Both the topology optimization technique and the principle of minimum strain energy are incorporated to find out the optimal configuration of the strut-and-tie model, which makes the model rational based. A design procedure for dimensioning and detailing the model is then given according to ACI 318-08 provisions. The proposed model is further evaluated using three test results, and the comparisons show that the model predictions are consistent and conservative. Moreover, a design example is presented to demonstrate the effectiveness and convenience of the proposed method.

On Saint-Venant’s Principle for Elasto-Plastic Bodies

This paper concerns Zanaboni’s version of Saint-Venant’s principle, which states that an elongated body in equilibrium subject to a self-equilibrated load on a small part of its smooth but otherwise arbitrary surface, possesses a stored energy that in regions of the body remote from the load surface decreases with increasing distance from the load surface. We here prove this formulation of Saint-Venant’s principle for elastic-plastic bodies. The present proof, which for linear elasticity considerably simplifies that developed by Zanaboni, depends crucially upon the principle of minimum strain energy to obtain a fundamental inequality that leads to the required result. Differential inequalities are not involved. The conclusion is not restricted to cylinders but is valid for plastic bodies of general geometries. Although no conditions are imposed on the plastic theories discussed here, counter-examples indicate that in certain circumstances the fundamental inequality, and hence the result, may be valid only for restricted data that includes the body’s minimum length.

Elliptical inhomogeneity in orthotropic composite materials due to uniform eigenstrains

Analytic solutions are presented for elastic fields induced by normal and shear eigenstrains in an elliptical region (inhomogeneity) embedded in orthotropic composite materials (matrix). Conformal transformation and complex function method are adopted to obtain exact stress distributions in the matrix at the interior boundary to calculate strain energy in the matrix. The stress fields in the elliptical inhomogeneity are derived analytically based on application of the principle of minimum strain energy of the elastic system to determine two fundamental variables characterizing the equilibrium boundary. The resulting solutions are verified with the continuity conditions for the normal and shear stresses along the interior boundary of the matrix. These reduce to known results for some special cases. Numerical examples are provided to illustrate the resulting solutions to the analysis of stress distribution pattern, strength-based failure and fracture behavior. The proposed model can be applied to evaluate volumetric expansion induced cracking and failure of polymers and polymer composites due to freezing of trapped moisture in the elliptical inhomogeneity region as well as fatigue strength of the materials under freeze–thaw conditions.

Modeling of failure due to freezing of moisture and freeze-thaw cycling in composite materials

Based on Eshelby's elastic inclusion model, an analytical model is developed for predicting cracking induced by volumetric expansion in orthotropic composite materials due to freezing of trapped moisture in a slender rectangular flaw region. Conformal transformation and complex function method are used to obtain stress distribution in the matrix at the interior boundary. The stress field in the rectangular inclusion is derived by solving for two important variables characterising the expanded equilibrium boundary determined by the principle of minimum strain energy. For this specific case, the analytical model predicts a crack driving force that far exceeds the fracture toughness of the material at that temperature. The model prediction is verified by comparison with actual test data. Besides predicting static fracture, it is demonstrated how the model can be adapted to predict fatigue life of composites under freeze-thaw conditions.

Failure in Composite Materials due to Volumetric Expansion of Freezing Moisture

A model is developed for predicting volumetric expansion induced cracking in orthotropic composite materials due to freezing of trapped moisture in a slender rectangular flaw region. Conformal transformation and the complex function method are used to obtain the stress distribution in the matrix at the interior boundary. The stress field in the rectangular inclusion is derived by solving for two important variables characterizing the expanded equilibrium boundary determined by the principle of minimum strain energy. The compressive stress acting on the long side of the rectangular inclusion acts as a crack driving force., The model is used to predict the occurrence of cracking due to volumetric expansion of ice in a specific composite, EXTREN, that has been observed in experiments. The model can be adapted to predict fatigue life of composites under freeze-thaw conditions.

Crack Driving Force in Anisotropic Media due to Non-Elastic Deformation

In this paper elastic stress field in an elliptic inhomogeneity embedded in orthotropic media due to non-elastic deformation is determined by the complex function method and the principle of minimum strain energy. Two complex parameters are expressed in a general form, which covers all characterizations of the degree of anisotropy for any ideal orthotropic elastic body. The stress acting on the long side of ellipse can be considered as a crack driving force and applied in failure and fatigue analysis of composites. For some special cases, the resulting solutions will reduce to the known results.

Matrix Cracking and Delaminations in Orthotropic Laminates Subjected to Freeze-Thaw: Model Development

With the increasing use of fibre composites in applications such as cryogenic liquid hydrogen tanks and repair/retrofitting of bridges, the diffusion and freezing of moisture to form ice is an issue of growing importance. The volumetric expansion of water when it freezes to form ice results in stress concentrations at the inclusion tip that may synergistically interact with the residual tensile stresses in a laminate at low temperatures to initiate a crack. In addition, understanding the long-term effect of daily and/or seasonal freeze-thaw cycling on crack growth in a laminate is of vital importance for structural durability. The objective of this paper is to establish a theoretical framework for the calculation of the stress intensity factor (KI) of a pre-existing crack in a composite structure due to the phase transition of trapped moisture. The constrained volume expansion of trapped moisture due to freezing is postulated to be the crack driving force. The principle of minimum strain energy is employed to calculate the elastic field within an orthotropic laminate containing an idealized elliptical elastic inclusion in the form of ice. It is postulated that a slender elliptical elastic inclusion can be used to approximate the stress field at the crack face, which can subsequently be used to calculate the stress intensity factor, KI, for the crack. The verification of the analytical model predictions and some potential applications will be published in a separate paper.

A refined analysis of the effective elasticity tensor for general cellular sandwich cores

The aim of the present study is the determination of the components of the effective elasticity tensor for two-dimensional cellular sandwich cores in consideration of core face sheet constraints. The microstructure is homogenized by means of a strain-energy based RVE concept assuming that strain states, which are equivalent on the macroscopic level, lead to equal strain energy in a representative volume element whether the real microstructure or the quasi-homogeneous “effective” medium is considered. The strain energy can be evaluated analytically if the cellular structure is decomposed into the individual cell wall elements, and assumptions are made for the displacement field of each cell wall. The displacement field of the core is approximated by a weighted superposition of the displacement field of the unconstrained core and an extension of the displacements of the face sheets into the core region. Since the approach is based on a kinematically admissible strain field in conjunction with the principle of minimum strain energy, the results provide rigorous Voigt type bounds for the effective normal and shear moduli. In general, a good agreement of the analytical results and the results of a finite element analysis is observed.

K-0914 複合圧力容器の網目補強(J05-1 FRP構造物物性)(J05 複合材料の加工と評価)

A new concept of structural reinforcement for compressed natural gas vehicle fuel containers is presented with formulation of structural optimization. The reinforcement is done by net, whereas the conventional composite containers are fabricated with continuous filament winding. The production efficiency is aimed at by the net reinforcement with high structural reliability ensured by means of structural optimization technique. The net is modeled as truss structure contacting with the surface of the metallic liner, which is modeled as elastic solid finite elements. The geometrical nonlinearity with the finite rotation of net members raises the shape finding of the net. The position of connection node is analyzed on the basis of principle of minimum strain energy with the iterative solution finding. The cylinder part of the container is relatively simple structure, so that the net reinforcement of spherical dome is taken into consideration. The optimum length of net member is determined so as to realize uniform strain net. A formulation of the sensitivity analysis is derived to constitute the governing equation of the optimum length, which is summarized in linear simultaneous equations. An optimum configuration of the net is searched for through the numerical example.

Elastic fields in polycrystalline or composite materials with slip inclusions

Inclusion slip induced by eigenstrains in polycrystalline or composite materials is modeled by adding Somigliana dislocations along the matrix-inclusion interface. It is shown that the elastic strain energy of a slip inclusion is in the minimum state. Therefore, a general and simple method to determine the slip represented by the Burgers vector is developed through the principle of minimum strain energy. The corresponding surface dislocation density is then obtained analytically. Following the eigenstrain formulation, the associated elastic fields inside and outside a slip spherical inclusion are presented in closed form. The results show that strains and stresses in the slip inclusion are not uniform, in contrast to the perfectly bonded inclusions where Eshelby's solution holds. The results of this work can be readily utilized in the solution of numerous problems pertaining to slip such as grain boundary sliding in polycrystals and granular media, behavior of precipitates at high temperature, and debonding of fibers in composite materials.

Recent Results on the Elasticity Theory of Inclusions

A method drawing from variational method is presented for the purpose of investigating the behavior of inclusions and inhomogeneities embedded in composite materials. The extended Hamilton’s principle is applied to solve many problems pertaining to composite materials such as constitutive equations, fracture mechanics, dislocation theory, overall elastic moduli, work hardening and sliding inclusions. Especially, elastic fields of sliding inclusions and workhardening rate of composite materials are presented in closed forms. For sliding inclusion problems, the sliding is modeled by adding the Somigliana dislocations along a matrix-inclusion interface. Exact formula are exploited for both Burgers vector and the disturbances in stress and strain due to sliding. The resulting expressions are obtained by utilizing the principle of minimum strain energy. Finally, explicit expressions are obtained for work-hardening rate of composite materials. It is verified that the work-hardening rate and yielding stress are independent on the size of inclusions but are dependent on the shape and the volume fraction of inclusions.

Effect of Thermal Stresses on the Vibration of Composite Cantilevered Plates

This study explores the effects of thermal stresses on the dynamic characteristics of composite cantilevered plates. The induced thermal stresses are determined by appealing to the principle of minimum strain energy, while the Raleigh-Ritz procedure is used to obtain the plate frequencies. Extensive numerical calculations have been carried out in order to gain quantitative understanding of how different choices of aspect ratios (AR) and temperature change intensities (T0) influence the natural frequencies of composite cantilevered plates. It has been shown that the first bending mode frequencies are unaffected by the induced stresses, while the first torsional mode frequencies decrease quite significantly with increasing values of T0, and therefore with increased magnitudes of the thermal stresses.