Small Strain Assumption Research Articles

A simple extension of Timoshenko beam model to describe dissipation in cementitious elements

In this paper, an extension of the Timoshenko model for plane beams is outlined, with the aim of describing, under the assumption of small displacements and strains, a class of dissipative mechanisms observed in cementitious materials. In the spirit of micromorphic continua, the modified beam model includes a novel kinematic descriptor, conceived as an average sliding relevant to a density of micro-cracks not varying along time. For the pairs of rough surfaces, in which such a distribution of micro-cracks is articulated, both an elastic deformation and a frictional dissipation are considered, similarly to what occurs for the fingers of the joints having a tooth saw profile. The system of governing differential equations, of the second order, is provided by a variational approach, endowed by standard boundary conditions. To this purpose, a generalized version of the principle of virtual work is used, in the spirit of Hamilton–Rayleigh approach, including as contributions: (i) the variation of the inner elastic energy, generated by the linear elasticity of the sound material and, in a nonlinear way, by the mutual, reversible deformation of the asperities inside the micro-cracks; (ii) the virtual work of the external actions consistent with the beam model, i.e., the distributed transversal forces and the moments per unit lengths; besides these two contributions, constituting the conservative part of the system, (iii) the dissipation due to friction specified through a smooth Rayleigh potential, entering a nonlinear dependence of viscous and Coulomb type on the sliding rate. Through a COMSOL Multiphysics implementation, 1D finite element analyses are carried out to simulate structural elements subjected to three- and four-point bending tests with alternating loading cycles. The dissipation of energy is investigated at varying the model parameters, and the predictions turn out to be in agreement with preliminary data from an experimental campaign. The present approach is expected to provide a valuable tool for the quantitative and comparative assessment of the hysteresis cycles, favoring the robust design of cementitious materials.

Analytical Formulation and Optimization of the Initial Morphology of Double-Layer Cable Truss Flexible Photovoltaic Supports

With the rapid development of the photovoltaic industry, flexible photovoltaic supports are increasingly widely used. Parameters such as the deflection, span, and cross-sectional dimensions of cables are important factors affecting their mechanical and economic performance. Therefore, in order to reduce steel consumption and cost and improve application value, it is crucial to design and optimize their initial morphology. In this paper, the mechanical behavior of a single-cable structure is introduced, and the simplified analytical formulations for internal force and displacement are deduced based on the geometric nonlinear characteristics and small strain assumption of the flexible photovoltaic supports. On this basis, the analytical expressions for the cable force and displacement of a convex prestressed double-layer cable truss flexible photovoltaic support structure under a uniform load are derived, and the correctness of the analytical formulations is verified by comparing the values with the finite element analysis results. In order to reduce the construction costs of the flexible photovoltaic support, a mathematical model for optimizing the initial structure’s morphology is established according to the analytical formulations. The initial morphology of the double-layer cable truss flexible photovoltaic support is optimized, and the optimization results of different deflection deformation limits and whether the lower load-bearing cable is allowed to relax are compared. The results indicate that the errors of the displacement formulation and cable force formulation, when compared with the finite element results, are less than 3% and 4%, respectively, which verifies the accuracy of the analytical formulations. By analyzing the cable force and displacement of the structure under static action, it is suggested that the deflection limit of the double-layer cable truss structure should be 1/100 of the single span. The lower load-bearing cables of the double-layer cable truss flexible photovoltaic support are highly susceptible to relaxation under wind suction loads, and, by comparing the optimization results, it is suggested that slack should be allowed in the lower load-bearing cables for a better economic effect. When choosing the most economical structure morphology, it is recommended that the total height of the mid-span struts should be 1/20~1/15 of the single span. The analytical formulation and the mathematical model for the optimization of the initial morphology proposed in this paper can provide certain theoretical references and bases for the design of practical engineering projects and play an important role in promoting its application and promotion.

Topology optimization of curved thick shells using level set method and non-conforming multi-patch isogeometric analysis

We present a novel framework for topological shape optimization of curved non-conforming multi-patch and trimmed thick-shells subjected to external loads. Our method integrates the level set method (LSM) with a diffuse interface, a Hadamard shape derivative, and multi-patch isogeometric analysis (IGA) into a gradient descent algorithm to systematically capture the evolution of the shape. This integration enables us to directly manipulate CAD-compatible geometries and analysis techniques and to obtain the results as a CAD surface. The novelty lies in the utilization of multi-patch IGA models based on NURBS functions, which allows us to simultaneously maximize the stiffness and minimize the volume of the shell by searching for an optimal material distribution within its middle surface. The material is modeled under a small strain assumption in linear elasticity using a Reissner–Mindlin kinematic shell model in plane stress. The effectiveness of our approach is demonstrated on several curved conforming and non-conforming multi-patch geometries in 3D.

A multiscale micromechanical progressive elastic-damage model for cementitious composites featuring superabsorbent polymer (SAP)

A multiscale micromechanics-based progressive damage model is developed to investigate the overall mechanical behavior and the interfacial microcrack evolutions of the cementitious composites featuring superabsorbent polymer (SAP) under uniaxial tension. Elastic properties, progressive damage process, and homogenization procedure of cementitious composites are systematically integrated in this model. The effective elastic moduli of the composites are determined based on a multiscale micromechanical framework. According to the small strain assumption, the total strain tensor and the elastic-damage compliance tensor are additively decomposed into elastic and damage-induced components. The damage-induced strains and compliances are then deduced from micromechanics. To characterize the progressive elastic-damage induced by microcracks, stages of microcrack propagation are identified from the interface contact stress and the matrix cleavage stress. The complex potentials and stress intensity factors for kinked interface cracks are derived from the distributed dislocations method. By implementing the homogenization process, the macroscopic mechanical behavior is obtained from the micro/mesoscale. The results indicate that the material parameters have clear mechanical significance. Different parameters, such as the SAP addition ratio, aggregate content, initial interfacial crack size, and initial interfacial crack location, are revealed to be influential in the overall mechanical behavior of the composites. The proposed model can be generalized to other particle-reinforced composites with different constituent properties, which can potentially contribute to the design and optimization of durable composites.

Equivalent linearization of hyperelastic rubber rings in dynamic vibration absorber boring bar and simulation of its dynamic properties

A slight perturbation in high-speed precision cutting could cause violent vibration of boring bars, resulting in unexpected wear and failure of the structures. In this paper, the dynamical characteristic of a boring bar with a dynamic vibration absorber (DVA) is investigated. Considering that the nonlinear characteristics of the rubber ring play a significant role in improving the capacity of DVA, we proposed an equivalent linearization method for modeling hyperelastic rubber rings based on the experimental data of uniaxial tensile test. The relationship between the stiffness of rubber rings and the precompression parameter δ is established for two types of rubber. The equivalent linearization model shows good accuracy and calculation efficiency while the mass of the DVA is confined to a narrow space and the assumption of small strain is inherently satisfied. Numerical results show that the time cost of the equivalent linearization model is only 5% of the full hyperelastic model. Furthermore, the effect of different spring stiffnesses and damping on the dynamic characteristics of the boring bar is compared by using the equivalent linearization model, and the vibration peak of the DVA boring bar is found to be reduced about 45%. Our findings establish a simplified modeling scheme for choosing rubber rings in design of dynamic vibration absorbers with acceptable accuracy and little time cost, which could also give some reference for modeling of metal rubber and gel-like materials.

Numerical simulation of miscible multiphase flow and fluid–fluid interaction in deformable porous media

AbstractThe focus of the underlying research work is on the macroscopic modeling of unstable multiphase fluid flow in deformable porous media, where a lower‐viscous fluid is displaced by a more viscous fluid. This process leads to the formation of channel‐like networks, called viscous fingering. The instability effect is involved in a wide range of different fields in engineering. Some of the most common applications include carbon sequestration to store carbon dioxide (CO2) in underground reservoirs, contaminant transport in geostructures, in industrial processes, such as filtration, catalytic reactions, and in the operation of polymer electrolyte membrane fuel cells with multiphase flow in the gas diffusion layers. In this work, ideal miscible water–glycerin fluids are considered. It is assumed that the interacting fluids in the deformable porous media are incompressible. In addition, a dispersion–diffusion law is applied to capture the fluid–fluid interactions. The presence of a deformable porous material adds additional effects to the problem of the multiphase flow in porous media. The stresses in the porous solid matrix lead to changes in the porosity and influence the flow velocity of the fluids. To couple the deformable porous media with the multiphase flow, a macroscopic approach is used that relies on the theory of porous media. For the porous solid matrix, a linear elastic material model within small strains assumption is applied. The influence of the deformation‐dependent porosity on the instability is studied for 2D simulations, such as a multilayered geometry with different elastic parameters. The presented coupled nonlinear system of differential equations is simulated with the finite element method. Furthermore, a stabilization technique based on the quasi‐compressibility method is used.

Three-dimensional nonlocal models of deformable ferroelectrics: A thermodynamically consistent approach

Within the framework of continuum thermodynamics, the paper develops a scheme for the study of piezo-ferroelectric materials undergoing large deformations. The ferroelectric polarization is decomposed into the sum of a reversible and a residual part which is considered as an independent variable. The modeling of the constitutive properties is made simpler by using referential, Euclidean invariant quantities. Constitutive functions depend on a set of variables that includes their time derivatives and gradients, and therefore must be consistent with a nonlocal statement of the second law of thermodynamics where the entropy production is represented as the sum of a non-negative supply and a flux. Both terms are also assigned by means of constitutive functions. A new and quite general model relating the residual-polarization rate and the ‘electric Gibbs free entropy’ is established. This thermodynamic potential is modeled according to the Landau-Devonshire approach and appropriate explicit expressions are proposed for anisotropic materials. Consequently, a new explicit evolution equation for the polarization vector is obtained for both high and low temperatures. In particular, the Landau-Devonshire scalar model is recovered in the isotropic case. Under the assumption of small strains and small polarization gradients, hysteresis and electromechanical coupling are described for a simple one-dimensional model. The advantage of our original approach is that a few constitutive parameters provide a good fit of material behavior.

Optimization design of a novel zero Poisson’s ratio honeycomb structure for in-plane mechanical performance

The excellent deformation performance of structure is the material basis of intelligent morphing aircraft. In this study, a zero Poisson’s ratio honeycomb structure consisting of sinusoidal corrugated walls is applied to morphing aircraft. The in-plane mechanical properties of the structure are deduced theoretically based on the assumption of small strain. According to the application requirements of tensile deformation and shear deformation, an optimization model is established with geometric parameters as optimization variables, and its mechanical properties are optimized by interior point method. The results show that the equivalent mechanical properties of the new zero-Poisson ratio honeycomb structure are greatly improved. The research provides important reference for the innovative design and optimization of honeycomb structures.

Rigorous solution for drained analysis of spherical cavity expansion in soils of finite radial extent

This paper revisits the expansion problem of a spherical cavity in an elastic–plastic soil mass with a finite radial extent under drained conditions, which proves to be non-self-similar. Rigorous semi-analytical solutions for the analysis of this problem are developed adopting several well-known soil models. With the small strain assumption, analytical expressions for both elastic stresses and displacements are obtained considering the stress dependency of soil moduli and influences of the outer boundary. In the plastic zone, the non-self-similar expansion process is formulated into a set of first-order partial differential equations (PDEs) based on the innovative combination use of the Lagrangian and Eulerian descriptions, and an efficient algorithm for solving these equations is developed. The solution method is quite general to be extended to many other constitutive models and is validated by comparing with existing solutions in some special cases. Then the cavity expansion behaviours in both clay and sand are discussed with a particular focus on the boundary effect. It is shown that, when the ratio of the outer to inner radius is small, the boundary effect may significantly influence the cavity expansion response. The present solution may provide a useful theoretical method for the validation of relevant finite element calculations and interpretation of some typical geotechnical problems such as cone penetration tests in small-size chambers, considering the boundary effect.

Effects of Mass Attachments on Flutter Characteristics of Thin-Walled Panels

Mass attachments may exist in the design and use of an aircraft panel, such as sensor layout, internal wiring, surface icing, etc. These mass attachments can change the flutter characteristics of the panel in supersonic flight and have important impacts on structural safety. In order to investigate the flutter characteristics of the panel with mass attachments, an assumed mode method is proposed to deal with the changes in the modal properties of the panel structure. Combined with the first order piston theory and p-k method, the flutter velocities and flutter frequencies of the panel under different cases can be obtained in the frequency domain. Firstly, based on the large displacement with a small strain assumption proposed by von Kármán and the proposed assumed mode method, the structural dynamic model of a simply supported panel with mass attachments and artificial dampers is constructed. Then, modal aerodynamic forces of the simply supported panel can be obtained based on first-order piston theory. Finally, flutter equations are transformed into the frequency domain and solved by the p-k method. The results showed that the existence of mass attachments can significantly change the flutter velocities and flutter frequencies of the panel. However, the flutter characteristics of the panel can be enhanced or recovered through some appropriate damper configuration schemes. Calculating the flutter characteristics of thin-walled panels with mass attachments can more accurately simulate real situations during flight, and one can obtain a safer design scheme of thin-walled panels.

Phase field microelasticity accommodating large deformation and modeling of voids evolution under creep

The phase-field microelasticity theory of Khachaturyan has been widely applied in materials science that is based on the small strain assumption. Here we develop an incremental realization of inelastic deformation (IRID) algorithm to deal with finite/large deformation with yet small elastic strain. A large deformation process is decomposed into a sequence of small deformation processes with intervals. At each interval the grids are stretched to accommodate the average inelastic deformation and advection/rotation operations are performed to account for the heterogeneous inelastic deformation. It is shown that Kachaturyan’s Fourier-transform based solutions can be adapted to the stretched grids. The IRID algorithm is compared against literature results regarding growth of a single void under remote stress of different triaxialities. Based on the IRID algorithm a phase-field model is developed that incorporates material microstructure, plasticity, multicomponent diffusion, and surface and grain boundary diffusion. The void growth kinetics and morphology evolution under coupling of diffusion and plasticity under creep are studied and the results show that coupling of the two mechanisms can significantly accelerate growth and coalescence of multiple voids at grain boundaries.

Thermal buckling analysis of thin-walled closed section FG beam-type structures

In this paper, thermal buckling analysis of thin-walled, functionally graded (FG) beam-type structures with closed cross-sections and various boundary conditions is investigated. One dimensional finite element model, based on Euler–Bernoulli–Vlasov​ theory, is employed under the assumptions of large rotations and small strains. Two cases of temperature rise across the thickness of the cross-section walls are considered, which are uniform and linear. Stability analysis has been performed in eigenvalue manner and in load–deflection manner using Newton–Raphson method. Numerical results for thin-walled box beams and trapezoidal beams for two different FG materials configurations are presented to investigate the effects of temperature distribution, boundary conditions, and material properties on the critical buckling temperature and global buckling behaviour. The accuracy and reliability of the beam model is verified by comparing it with several benchmark examples. Good agreement was observed, showing possibility of further application of the proposed model which requires reduced computational time and provides greater modelling flexibility compared to commercial codes.

Stress gradient plasticity theory development for cyclic and monotonic loading of thin-walled structures: Back-stress, size effect, passivation effect

Small scale plasticity behaviour is modelled by strain or stress gradient plasticity models. The stress gradient plasticity theory, proposed by Chakravarthy and Curtin, is a constitutive consideration of the stress gradient that affects the initial flow as well as strain hardening rate. We apply the lower-order stress gradient plasticity model to different monotonic problems and cyclic torsion, under small strain and rotation assumption. The stress gradient can be due to the problem geometry and loading and/or due to passivation. Passivation is considered in different monotonic loadings of classical problems. To use the stress gradient plasticity model in cyclic loading problems, a back-stress model is used and the kinematic hardening is considered. The developed model in microscale plasticity considers dislocation pile-ups, grain size, and dislocation density. The Bauschinger effect is also considered during reverse loading by the consideration of the kinematic hardening. The cyclic torsion of thin copper wires of different diameters is modelled by the developed model. Comparisons with the experimental results and distortion strain gradient plasticity model are also performed. It is shown that the simple developed model can capture the essential microscale plasticity behaviour of metals, with only one length scale parameter.

Multiaxial Damage Ratio Strength Criteria for Fiber-Reinforced Concrete

Based on the damage ratio strength theory, a dimensionless damage ratio strength criterion for fiber-reinforced concrete (FRC) under true triaxial stress states was established under the assumption of small strains. Based on the experimental data of steel fiber–reinforced concrete (SFRC), polypropylene fiber–reinforced concrete (PFRC), hybrid fiber–reinforced concrete (HFRC), and steel fiber–reinforced high-performance lightweight concrete (SFRHLC), parameters for the damage ratio criterion were recommended. The damage ratio values were verified by SFRC experimental results under uniaxial, biaxial, and triaxial loading conditions, and the damage ratio values of SFRC under uniaxial tension, uniaxial compression, and biaxial equal compression stress conditions were compared with those of plain concrete. The true triaxial damage ratio strength criterion can be reduced to the conventional triaxial criterion and biaxial criterion, and the corresponding simplified strength criteria were proposed. The proposed dimensionless six-parameter criterion agrees well with the experimental results of the aforementioned materials for true triaxial, conventional triaxial, and biaxial loading conditions. Compared with the strength criteria in other studies, the proposed criterion can accurately represent the strength characteristics of FRC.

Damage ratio strength criterion for cast iron

According to the assumption of small strains and total strain decomposition of cast iron, the expressions of the inelastic principal strain rate and relative energy dissipation rate of cast iron are derived, and the parameters of the tensile and compressive damage ratio of cast iron in the inelastic phase are proposed. Based on the least energy consumption principle, combined with the above expressions and parameters, the damage ratio strength criterion for cast iron under biaxial stress states is established. The damage ratio variable expression of the cast iron damage ratio strength criterion considering the effect of Lode angle is proposed. Furthermore, the values of the damage ratio variable are verified by the experimental stress–strain curves of cast iron under uniaxial stress states. By comparing the predicted values to 104 experimental data points of cast iron specimens, the failure envelopes of the proposed criterion can represent the failure behavior of cast iron under biaxial loading conditions. To facilitate the practical application in engineering, the further simplified expression of the four-parameter damage ratio strength criterion for cast iron under biaxial stress states is suggested and those empirical parameters are recommended. Compared with some main strength criteria, the simplified criterion is in better agreement with the experimental results under every stress state.

Thermal–Structural Coupling Analysis for Multiple Honeycomb Plates Connected by Hinges

The dynamical model of multiple honeycomb plates connected by hinges, including the thermal–structural coupling effect, is investigated in this paper. Under the assumption of small deformations and strains, the classical plate theory is used to establish the dynamical model of the honeycomb plate. The Chebyshev polynomial is adopted to describe the deformation of the plate and the linear torsional spring to represent the elastic hinge. Using the Rayleigh–Ritz method, the characteristic equation of the honeycomb sandwich structure is derived, from which the natural frequencies are obtained. Considering the thermal effect of the plate, the thermal–structural coupling dynamical equation is obtained. The thermal conduction in the thickness direction is calculated by the finite difference method. The present model is validated by comparing with the result generated by an equivalent orthotropic plate model established by the finite element (FE) software MSC Patran. The response analysis reveals the important effect of the metal skin and the hinge stiffness on the dynamic characteristics of the honeycomb sandwich structure. Numerical simulations reveal thermal–structural coupling features for the multi-honeycomb-plate structure from the perspective of the change in dynamic characteristics and the distribution of the temperature gradient. The thermal–structural coupling calculation method adopted herein can be used to solve the thermal conduction of the plate. It serves as the theoretical method for analyzing the dynamical behavior of the solar panel subjected to the solar heat.

Phase-field modeling of fracture in strain-hardening elastomers: Variational formulation, multiaxial experiments and validation

Elastomeric sealing and rubber-like materials find applications in a variety of industrial domains. Understanding the behavior of such materials and description of fracture in elastomeric components is therefore decisive in the choice of materials for realizing newer concepts and product designs. Over the years, phase-field modeling of fracture has proven to be a capable tool in handling arbitrary cracks and their propagation in an adequate manner. Nevertheless, a large number of associated models are based either on small-strain assumptions or focus on phenomenological hyperelastic models, both of which are seldom of interest to industrial-grade rubbers. In this work, we present a way to perform phase-field fracture simulation in conjunction with the renowned Gent model of hyperelasticity, wherein the singularity issue near the limit of chain stretch is handled adequately. An extension of an existing model is proposed to describe crack propagation in multiaxial loading cases. The salient features of the model are demonstrated by suitable numerical examples. Furthermore, we perform material characterization experiments on an industrially relevant silicone material and use the above framework to determine the model parameters of this elastomeric adhesive. Additionally, the ability of the model to predict stress–strain response and crack path in different deformation modes is demonstrated.

Geometrically Nonlinear Analysis of Laminated Composite Plates Using Cell- and Edge-Based Smoothing MITC3 Finite Elements

A novel finite element model for geometrically nonlinear analysis of laminated composite plates is proposed in this paper. It is based on a seven-variables higher-order shear deformation theory that only requires a C0-type continuity. The finite element method with mixed interpolation of tensorial components approach (MITC) is considered in which an MITC3 three-node triangle element with seven degree-of-freedoms per nodes, cell- and edge-based strain smoothing techniques is developed. The geometrical nonlinearity of the plates involving by von Karman strain is formed in the total Lagrange approach based on the small strain assumptions. Numerical examples of cross-ply laminated composite plates are provided to investigate the validation of the proposed method.

Finite element simulation of quasi-static tensile fracture in nonlinear strain-limiting solids with the phase-field approach

We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed.

Experimental study and numerical modeling of the thermo-hydro-mechanical processes in soil freezing with different frost penetration directions

This research work presents an experimental and numerical study of the coupled thermo-hydro-mechanical (THM) processes that occur during soil freezing. With focusing on the artificial ground freezing (AGF) technology, a new testing device is built, which considers a variety of AGF-related boundary conditions and different freezing directions. In the conducted experiments, a distinction is made between two thermal states: (1) The thermal transient state, which is associated with ice penetration, small deformations, and insignificant water suction. (2) The thermal (quasi-) steady state, which has a much longer duration and is associated with significant ice lens formation due to water suction. In the numerical modeling, a special focus is laid on the processes that occur during the thermal transient state. Besides, a demonstration of the micro-cryo-suction mechanism and its realization in the continuum model through a phenomenological retention-curve-like formulation is presented. This allows modeling the ice lens formation and the stiffness degradation observed in the experiments. Assuming a fully saturated soil as a biphasic porous material, a phase-change THM approach is applied in the numerical modeling. The governing equations are based on the continuum mechanical theory of porous media (TPM) extended by the phase-field modeling (PFM) approach. The model proceeds from a small-strain assumption, whereas the pore fluid can be found in liquid water or solid ice state with a unified kinematics treatment of both states. Comparisons with the experimental data demonstrate the ability and usefulness of the considered model in describing the freezing of saturated soils.