Incremental Equilibrium Equations Research Articles

A numerical approach to the stability analysis of growing cylindrical tubes and spherical tissues

In this paper, the effect of growth on the stability of elastic materials is examined through a numerical approach. Growth and resorption are considered to have two main effects from the stability standpoint. Corresponding to the change in mass, the geometry of a system changes, and the critical length of the system can be modified. In addition, the growth of the material may be affected by the stress, yet it may also impose residual stresses. As a result, the material is either stabilized or destabilized by these stresses. As a general framework for the description of elastic properties, the theory of finite elasticity is employed to investigate the growth of elastic materials. Growth is taken into account through matrix multiplication of the deformation gradient. The formalism of incremental deformation is adopted to consider growth effects. Using this formalism, the stability of growing neo-Hookean incompressible cylindrical and spherical shells under external pressure is considered. Firstly, the incremental equilibrium equations and the corresponding boundary conditions for the incompressible growing shells are summarized and afterward employed in order to analyze the behavior of spherical and cylindrical shells subjected to external pressure. The generalized differential quadrature method (GDQ) is utilized to solve the eigenvalue problem that results from a linear bifurcation analysis. It is shown that this numerical method can be efficiently utilized to solve the considered problem. The results are in full agreement with the previously obtained results. In the presence of external radial pressure, an elastic shell will buckle circumferentially to a noncircular cross-section. A change in thickness due to the growth can significantly affect buckling, both in terms of the critical pressure and the buckling pattern. Finally, the effects of the thickness ratio A1/A2, mode number n, and the growth parameter γ are studied on the shell’s stability.

Form-finding and shape adjustment of cable-membrane reflectors

Cable-membrane reflector is a highly nonlinear flexible structure, which can represent a specific surface shape under the proper combination of cable tension and membrane stress. AstroMesh reflector is a representative of cable membrane reflector, which has advantages of large aperture, easy folding, and light weight, and is increasingly utilized in space antenna reflectors. In this paper, a finite element model of the cable-membrane structure is established together with the corresponding incremental equilibrium equation which provides the sensitivity of the nodal position with respect to the geometric parameter of the cables and triangular membranes. A form-finding strategy is detailed, which aims at a configuration with the reflector nodes locating on a specified paraboloid while all cables and membranes are tensioned uniformly as possible. In addition, the surface precision of the reflector is reduced due to the inevitable errors in the manufacturing process. Therefore, the tension ties in the cable network is designed to be adjustable to solve this problem. Carefully changing the length of these cables can increase the reflector surface precision, which is referred as shape adjustment. An adjustment strategy with model updating is proposed. The model updating using the measured positions of the reflector nodes can dramatically reduce the discrepancy of the finite element model with respect to the actual reflector, thus remarkably improving the efficiency of shape adjustment. Numerical example and physical experiment demonstrate the effectiveness of the proposed method.

Penalty function method for geometrically nonlinear buckling analysis of imperfect truss with multi-freedom constraints based on mixed FEM

This paper is concerned with the approach to implementing the Penalty function method for imposing multi-freedom constraints in geometrically nonlinear analysis of imperfect trusses based on mixed finite element formulation. Using a finite element model based on displacement formulation, it is required to incorporate both the dependent boundary relations and initial length imperfection to the nonlinear master stiffness system of equations for solving the geometrically nonlinear problem of imperfect truss with multi-freedom constraints. For decreasing the mathematical complexion of the incorporating process, the author proposes a novel mixed finite truss element considering initial imperfection, used in building the model for solving the geometrically nonlinear problem of truss with multi-freedom constraints. The modified nonlinear stiffness equation is constructed by employing the penalty function method to convert a constrained problem into an unconstrained problem by extremizing the augmented energy function established based on the proposed mixed finite element formulation. For solving the nonlinear equilibrium equation of imperfect trusses with multi-freedom constraints, the incremental equilibrium equation is constructed, and the incremental-iterative algorithm for calculation is established utilizing the arc-length method, used for writing the calculation program for investigating geometrically nonlinear behavior of imperfect truss with multi-freedom constraints. The results of the numerical test show the influence of initial imperfection and choosing weight values on the equilibrium path of the truss.

On the Stability of Hyperelastic Spherical and Cylindrical Shells Subjected to External Pressure Using a Numerical Approach

In this paper, the incremental equilibrium equations and corresponding boundary conditions for the isotropic, hyperelastic and incompressible shells are derived and then employed in order to analyze the behavior of spherical and cylindrical shells subjected to external pressure. The generalized differential quadrature (GDQ) method is utilized to solve the eigenvalue problem that results from a linear bifurcation analysis. The results are in full agreement with the previously obtained results and the effects of thickness and mode number are studied on the shell’s stability. For the spherical and cylindrical shells of arbitrary thickness which are subjected to external hydrostatic pressure, the symmetrical buckling takes place at a value of [Formula: see text] which depends on the geometric parameter [Formula: see text] and the mode number [Formula: see text], where [Formula: see text] and [Formula: see text] are the undeformed inner and outer radii, respectively, and [Formula: see text] is the ratio of the deformed inner radius to the undeformed inner radius.

Stability analysis of shear deformable cross-ply laminated composite beam-type structures

This paper presents a shear deformable beam model for the nonlinear stability analysis of composite beam-type structures. Each wall of the cross-section is stacked in the cross-ply scheme. The incremental equilibrium equations are derived in the framework of an updated Lagrangian formulation. Hooke’s law is assumed to be valid. The nonlinear displacement field of the composite cross-section accounts for the restrained warping and the large rotation effects. The shear deformation effects are included through Timoshenko’s bending theory and modified Vlasov’s torsion theory. Additional shear correction factors due to the bending-torsion coupling occurring at asymmetric cross-sections are introduced in the analysis. The accuracy and reliability of proposed numerical mode are verified on benchmark examples, and the obtained results confirm that it can be classified as a shear locking-free model.

Analysis of nonlinear soil-structure interaction using partitioned method

For the current state-of-practice in soil-structure interaction (SSI) of nuclear facilities, the strain-dependent characteristics of soil are considered indirectly via equivalent linear methods in the frequency domain, typically represented by the SASSI program. SSI analysis in the time-domain, although directly involving material nonlinearity of the soil, is inefficient in practice up to now. Based on Partitioned Analysis of SSI (PASSI), a method for nonlinear soil-structure interaction analysis is developed in this paper, by applying implicit time-step integration and explicit one with different time steps to nuclear island and to soil, respectively. The incremental equilibrium equation and explicit decoupling method are used to analyze the soil nonlinearity described by the Davidenkov model with simplified loading-reloading rules, which avoids solving algebraic equations and iteration processes. An asynchronous parallel algorithm for nonlinear SSI analysis is given. A simple example is given to verify the partitioned approach with explicit-implicit co-computation against the full-explicit approach. Seismic response characteristics of nuclear power plants are investigated by comparing responses of nonlinear analysis with linear analysis. Different results address the importance of nonlinear SSI analysis for the seismic performance of nuclear structures.

Multi-Directional Shape Change Analysis of Biotensegrity Model Mimicking Human Spine Curvature

This paper presents a numerical strategy for the shape change analysis of spine biotensegrity models in multi-directional modes. The formulation of incremental equilibrium equations and optimization problem for shape change analysis via the forced elongation of cables to achieve the target coordinates of the monitored nodes of spine biotensegrity models are presented. The distance between the monitored nodes and the target coordinates is chosen as the objective function which is minimized subject to inequality constraints on member axial forces and cable forced elongation. Three spine biotensegrity models were analyzed to validate the effectiveness of the proposed method. The deformation characteristics of the Class-1 four-stage biotensegrity models mimicking the natural curvature of the human spine were investigated. A highly successful rate in achieving the target coordinates was observed in a total of 258 analysis cases, with percentages of 99.9%, 99.9% and 98.9% for shape change analysis involving uni-, bi- and tri-directional modes, respectively. The results show that the spine biotensegrity models have more flexibility in undergoing bending in comparison with axial deformation. With the established shape change strategy, the flexibility and versatility of the movement of spine biotensegrity models can be further studied for potential application in the shape change control of deployable structures together with the use of IoT.

Investigation on the performance of the expansion water seal by using isogeometric analysis with variational inequalities for the frictional contact

In this study, IGA is introduced into the performance analysis of the expansion water seal for an exact representation of its complex geometrical shape. Firstly, the incremental equilibrium equations used in the large deformation analysis are derived for the IGA discretization model based on the updated Lagrangian formulation. The unknown contact forces are directly involved in the equilibrium equations, which results in a simple expression. Then, through the investigation of the contact conditions, the normal and tangential contact forces are verified to be the solutions of two box-constrained variational inequalities representing the normal and tangential contact conditions, respectively. Furthermore, the incremental equilibrium equations for all components of the water seal system and the variational inequalities for all contacts between the components are assembled together and reformulated as a global variational inequality, which is resolved by using the Extra-gradient method. Ultimately, the new method is applied in the performance comparison of two optional expansion water seals, in which the capability and precision of the proposed method are investigated. Results verify that the proposed method is effective in the numerical simulation of the expansion water seal and has a higher precision than the traditional FEM under the same conditions.

Nonlinear stress analysis of shell structures in buckling and snapping problems by exact geometry solid-shell elements through sampling surfaces formulation

In this paper, the nonlinear three-dimensional (3D) stress analysis of shell structures in buckling and snapping problems is presented. The exact geometry or geometrically exact (GeX) hybrid-mixed four-node solid-shell element is developed using a sampling surfaces (SaS) method. The SaS formulation is based on the choice of N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. The SaS are located at the Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree N), that is, the outer surfaces are not included into a set of SaS. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the through-the-thickness distributions of displacements, strains and stresses leads to an efficient higher-order shell formulation. The incremental equilibrium equations are solved by the Newton–Raphson method combined with the Crisfield arc-length algorithm. The tangent stiffness matrix is evaluated by effective 3D analytical integration. As a result, the proposed GeX hybrid-mixed solid-shell element exhibits superior performance for coarse meshes and allows much larger load increments than is possible with existing displacement-based solid-shell elements. This can be useful for the 3D stress analysis of thin and thick shells in different states such as pre-buckling, bifurcation and post-buckling.

Stability of pre-stressed incompressible hyperelastic cylindrical tubes under axial compression

The problems of stability and controllability of the finite deformation are investigated for a pre-stressed hyperelastic cylindrical tube subjected to the axial compression. The tube is assumed to be composed of an incompressible neo-Hookean material. Based on the theory of small deformation superposed on large elastic deformation, the mathematical model is formulated by a system of incremental equilibrium equations and incremental boundary conditions. Moreover, the general solutions describing the finite deformation of the tube are obtained by the form of Bessel functions. Finally, the system of nonlinear equations governing the stability of the tube are given, the criteria of the stability discussed in terms of the corresponding numerical simulations.

Double control method for construction form finding of glass curtain system

An incremental equilibrium equation was deduced for the construction of glass curtain system with inverted-cone curved single-layer cable-net based on the theory of non-linear finite element method. In view of actual influence of glass installation during the construction process, analysis is made on the correction algorithm for formation of non-linear stiffness matrix of structure with consideration of glass installation. The developed algorithm allows for double control over the shape and pre-tension during the construction, and accurate determination of the structural configuration and cable force. Feasibility and accuracy of the method are verified and supported by the case study results.

Disclinations and GND tensor effects on the multislip flow rule in crystal plasticity

This paper deals with new elastoplastic models for crystalline materials with microstructural defects, such as dislocations and disclinations, which are consistent with the multislip plastic flow rule, and compatible with the free energy imbalance principle. The defect free energy function is a function of the disclination tensor and its gradient, and of the geometrically necessary dislocation (GND) tensor, via the Cartan torsion. By applying the free energy imbalance, the appropriate viscoplastic (diffusion-like) evolution equations are derived for shear plastic rates (in slip systems) and for the disclination tensor. The two sets of differential (or partial differential, i.e., non-local) equations describe the rate form of the adopted disclination–dislocation model. The first set is typical for finite deformation formalism, while the second set refers to the evolution equations with respect to the reference configuration. The dislocation appears to be a source for producing disclination defects. A pure dislocation elastoplastic model is also proposed. Multislip models with disclination within the small deformation approach are derived from the finite deformation models. The initial and boundary value problems are formulated and the incremental (rate) equilibrium equation leads to a variational equality for the velocity field, at any time, which is coupled with the rate type models for the set of variables. First, the elastic problem is solved for a certain time interval by assuming that the existing defects inside the body remain inactive. Subsequently, the variational equality is solved for the velocity field, at any time, if the slip systems are activated. Consequently, the state of the body with defects is defined by the solution of the differential-type equations, when the velocity field is known for a certain time interval. Appropriate initial conditions are necessary, including those associated with defects which became active. Finally, an update algorithm must be provided in order to compute the fields at the current moment.

Buckling loads of nano-beams in stress-driven nonlocal elasticity

Size-dependent buckling of compressed Bernoulli-Euler nano-beams is investigated by stress-driven nonlocal continuum mechanics. The nonlocal elastic strain is obtained by convoluting the stress field with a suitable smoothing kernel. Incremental equilibrium equations are established by a standard perturbation technique. Higher-order constitutive boundary conditions are naturally inferred by the stress-driven nonlocal integral convolution, equipped with the special bi-exponential kernel. Buckling loads of compressed nano-beams, with kinematic boundary constraints of applicative interest, are numerically calculated and compared with those obtained by the theory of strain gradient elasticity.

A shearable and thickness stretchable finite strain beam model for soft structures

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.

Surface instability of elastic half-spaces by using the energy method

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res. 12 , 168–182 ( doi:10.1007/BF03184638 )) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.

Numerical Investigation on the Buckling Failure of Slender Tubular Member with Cutout Presence

Pitting corrosion often leads to the creation of small holes in steel tubular member of platform structures when a protective coating is damaged. A single pit on slender compression element can cause a significant reduction in the buckling capacity of the member. Euler formula is no longer applicable for determining the critical buckling load when cutout presence on the member. This research was conducted to numerically study the effect of a circular hole on the buckling capacity of slender steel tubular member. A variation on hole positions was at 0.125 L, 0.25 L, 0.375 L, and 0.5 L, where L is the length of the member. The hole was taken to be 0.5 pipe diameter. Two nonlinear geometric 3D Finite Element models were developed to analyzed the member critical buckling load: (a) buckling analysis, where the problem was formulated as eigenvalue problem based on the nonlinear incremental equilibrium equations, and (b) nonlinear analysis, where the nonlinear equilibrium equations were developed and solved by several schemes to get the load – deflection curve. For the both models, the tubular member was discretized into: (a) shell elements, and (b) solid elements. The numerical results were verified by experimental investigation. The results showed that: (a) the presence of cutout reduced the buckling load significantly, (b) the reduction ranging from 3% to 10% depending on the hole positions, (c) the maximum reduction occurs when the hole position was in the middle of the member length, (d) compared to experimental results, the critical buckling load obtained from buckling analysis deviated 1~4% while those of nonlinear analysis deviated 1~5%, (e) the buckling mode corresponded with member bent away to opposite side of the cutout position.

On the Geometrically Nonlinear Elastic Response of Class θ = 1 Tensegrity Prisms

The present work studies the geometrically nonlinear response of class ϑ=1 tensegrity prisms modeled as a collection of elastic springs reacting in tension (strings or cables) or compression (bars), under uniform uniaxial loading. The incremental equilibrium equations of the structure are numerically solved through a path-following procedure, with the aim of modeling the mechanical behavior of the structure in the large displacement regime. Several numerical results are presented with reference to a variety of physical models, which use two different materials for the cables and the bars, and show different aspect ratios associated with either 'standard' or 'expanded' configurations. An experimental validation of the predicted constitutive response is conducted with reference to a 'thick' and a 'slender' model, observing rather good theory vs. experiment matching. The given numerical and experimental results highlight that the elastic response of the examined structures may switch from stiffening to softening, depending on the geometry of the system, the magnitude of the external load, and the applied prestress. The outcomes of the current study confirm previous literature results on the elastic response of minimal tensegrity prisms, and pave the way to the use of tensegrity systems as nonlinear spring units forming tunable mechanical metamaterials.

Commutativity of the strain energy density expression for the benefit of the FEM implementation of Koiter's initial postbuckling theory

SummaryThe concept of full commutativity of displacements in the expression for strain energy density for the geometrically nonlinear problem has been introduced for the first time and fully established in this paper. Its consequences for the FEM formulation have been demonstrated. As a result, the strain energy, equilibrium equation, and incremental equilibrium equation for the geometrically nonlinear problem can all be presented in a unified manner involving various stiffness matrices that are all symmetric, unique, and explicitly expressed. As an important application, the framework has been employed in the FEM implementation of Koiter's initial postbuckling theory, which has been handicapped by its mesh sensitivity in evaluating one of the initial postbuckling coefficients. This has largely prevented it from being incorporated in mainstream commercial FEM codes. Based on the outcomes of this paper, the mesh sensitivity problem has been completely resolved without the need to use any specially formulated element. As a result, Koiter's theory can be practically and straightforwardly implemented in any FEM code. The results have been verified against those found in the literature.

A shear-deformable beam model for stability analysis of orthotropic composite semi-rigid frames

In this paper, a shear deformable beam model for nonlinear stability analysis of frames made of composite materials is presented. Each wall of the cross-section is assumed to be orthotropic in such a way that normal stress does not cause shear strains to occur. The incremental equilibrium equations for a straight thin-walled beam element are derived within the framework of updated Lagrangian formulation and the nonlinear displacement field of cross-sections, which accounts for the restrained warping and the large rotations effects. Timoshenko’s theory for non-uniform bending and modified Vlasov’s theory for non-uniform torsion are applied to include the shear deformation effects. The coupled bending-torsion shear deformation effects occurring at the asymmetric cross-section are also included in the model. To account for the semi-rigid connection behaviour, the hybrid finite element is introduced through the special transformation procedure. Several benchmark examples are demonstrated for verification purposes. The obtained results indicate that the proposed model can be classified as shear locking-free one.

Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems

Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.