Elastic Part Research Articles

Stress evolution during the megathrust earthquake cycle and its role in triggering extensional deformation in subduction zones

Great (moment magnitude Mw ∼8.0 and larger) subduction megathrust earthquakes are commonly followed by increased rates of normal faulting seismicity. Extensional activity within the subducting slab is amplified when megathrust slip propagates close to the trench, and forearc extension is triggered by the largest magnitude (Mw 8.5 and larger) events. To better understand these observations, we develop an earthquake cycle model with a realistic slab geometry and stresses that are in balance with plate interface slip and bulk viscous relaxation. The modeled stresses represent perturbations to the long-term background tectonic stress field. The steady-state inter-seismic earthquake cycle stresses are compressive and their magnitudes depend on the interface locking configuration, from 25 MPa for a fully locked seismogenic zone to 5 MPa for discrete asperities on an otherwise unlocked plate interface. The co-seismic slip and corresponding co-seismic stress changes are similar in these models, independent of the locking configuration. The co-seismic stress change magnitudes are up to 5 MPa. This implies that the earthquake cycle stresses after the mainshock would still be compressive in the case of a continuous seismogenic zone, but would be widely reset to zero in the case of discrete asperities. Models with co-seismic slip confined to shallow depths produce tensional stress changes in the slab and the forearc near the trench, whereas events rupturing the base of the seismogenic zone produce tensional stress changes limited to region immediately surrounding the rupture. The locations of normal faulting aftershocks generally correlate well with the tensional co-seismic stress changes greater than 1 MPa. Following the mainshock, rapid afterslip occurring down-dip of the megathrust rupture expands the region of relative extension, but reduces its magnitude and does not promote additional normal faulting events. Bulk viscous relaxation does little to the state of stress in the elastic parts of the model. Continued plate convergence across a re-locked interface returns the system to the pre-earthquake state of compression.

АДИТИВНЕ ФОРМУВАННЯ ЕЛАСТИЧНИХ ВИРОБІВ З ПВХ ПЛАСТИЗОЛЮ

Development and testing of the technology of additive formation of elastic and soft products. The tensile strength and elongation at break for all the studied samples were determined according to ISO 527, the density of the samples according to PN-EN ISO 1183-1, the melt flow rate according to ISO 1133: 2005, the Shore A hardness according to ISO 868. The technology of additive manufacturing of soft and elastic products from liquid consumables has been created. The initial raw material for the formation of objects is the liquid composition of PVC, which can be transformed from the liquid state of plastisol to elastic plastic part or article. The created technology allows to create products of complex geometric shape with a hardness on the Shore scale of 5 to 90 units. The developed technology allows to create products from composite materials and foam based on PVC and can be implemented on equipment for FFF technology. The composition of the plastisol allows to adjust the hardness and elasticity of the products obtained over a wide range, the addition of pigments and dyes allows reach different colors. The developed technology has a higher forming speed rate than the FFF technology. A possible field of application is the manufacture of seals, soundproof materials, shock and vibration absorbers, design elements of clothing and shoes. The possibility of additive manufacturing of PVC plastisol on a fabric has been experimentally confirmed. The new features of the addition manufacturing of liquid PVC plastisol and properties of products obtained by the method are investigated. Product formation takes place through the process of simultaneously converting plastisol to molten plastic compound in an extruder. The method of additive manufacturing of products from liquid plastisol is developed, which allows to obtain products with low hardness and high elasticity. The developed technology allows the formation of products from composite materials and foams with the modernization of widely available equipment.

Modified flag-shaped model for self-centering system and its equivalent linearization and structural optimization for stochastic excitation

The analysis and design of self-centering structural systems have attracted substantial attention from researchers for the seismic design of structures. The flag-shaped hysteretic model has been widely used in the design and structural response analysis of self-centering devices. In this research, a modified flag-shaped (MFS) model is proposed to describe the hysteretic characteristics of self-centering energy dissipation (SCED) braces, which are commonly used in self-centering structures. The MFS model comprises three parts: a linear elastic part, a bilinear elastic part, and an elasto-plastic part with a slip zone. The effectiveness of the MFS model is validated based on the results of the quasi-static and hybrid simulation tests available in the literature. Equivalent linearization is carried out on the MFS model for stochastic earthquakes, and closed-form solutions are obtained for the linearized parameters. A one-degree-of-freedom braced system is used to verify the equivalent linearization of the MFS model in comparison with Monte Carlo simulation results. Finally, an SCED-braced five-story frame structure is optimized by minimizing the maximum ductility demand for the SCED braces using the equivalent linearization of the MFS model. The optimized design by the MFS model is observed to exhibit uniform ductility demands along the height. This can be considered as the ideal optimal solution. The responses of the multi-degree-of-freedom structure also demonstrate the effectiveness of the proposed MFS model and its equivalent linearization.

Shear force‐deformation relationship of concrete‐filled square steel tube beam‐columns under monotonic lateral loading

AbstractThis paper proposes a simplified multi‐linear model of a shear force‐deformation relationship for concrete‐filled square steel tube short to slender beam‐columns. To evaluate the behavior of short to slender beam‐columns with a single model, the column member is divided into elasto‐plastic part and an elastic part. For the elasto‐plastic part, the deformation at each point is calculated based on the bending moment‐curvature relation model proposed by the author and by employing a newly proposed evaluation formula for curvature magnification. Predictions from the model are found to agree approximately with the experimental results up to large deformations.

Hydrodynamic damping enhancement by implementing a novel combined rigid-elastic heave plate

Heave plates are structural components used for reducing the vibrations caused by environmental forces on marine and offshore structures by changing the hydrodynamic properties. The fact that the added mass increase via heave plates does not always lead to the structural response reduction underscores the role of damping in maintaining the vibration amplitude within allowable limits. In the present experimental study, a novel combined rigid-elastic design is used to improve the damping through the velocity increase in the elastic part and added mass creation in the central rigid part. The desired percentage of total added mass and damping can be adjusted by changing the rigid-to-elastic parts diameter ratio, which is the main scope of this experimental research. Frequency of vibration, which affects the elastic edge excited mode shapes, also affects the forming of the vortex shedding. Experimental tests show that the frequency increase generally causes high damping performance provided that the excited mode shapes are axisymmetric, which strongly depends on equivalent stiffness and mass.

Constitutive model of rock triaxial damage based on the rock strength statistics

Firstly, an X-ray diffraction test is carried out to investigate brittle rock specimens’ composition, and a triaxial compression test is conducted to study the deformation behaviors and mechanical properties. Then, assuming that the rock material is able to be divided into the elastic part satisfying the Hooke’s law and the damage part where rock strength follows lognormal distribution, this paper determines a damage variable and establishes a damage constitutive model which effectively reflects the residual strength in the process of rock failure. Meanwhile, testing data are used to validate the reliability of the proposed model in this paper and the dependence of statistic parameters on the confining pressure. Finally, impacts caused by statistic parameters variation on compression strength are analyzed comprehensively, and we also conduct a comparison between this proposed model and other models from other literatures, thereby showing the reliability and rationality of this proposed model. In addition, the research outcomes presented in this paper also can effectively offer significant reference for the development of rock mechanics and rock engineering.

High performance quasi-isotropic thin-ply carbon/glass hybrid composites with pseudo-ductile behaviour loaded off-axis

The aim of this work was to investigate the effect of loading angle variation on the pseudo-ductility of quasi-isotropic (QI) hybrid composite laminates. Previously, hybrids of thin-ply carbon fibres and standard glass fibres were found to have an excellent pseudo-ductile behaviour both in unidirectional (UD) and QI configurations when subjected to axial tension in the fibres’ orientations. In this work, the QI laminates, with 60° intervals, have been subjected to a quasi-static tensile load at various off-axis orientations – i.e. 5°, 10° and 20°. The QI hybrid composites were made by sandwiching a QI T300-carbon laminate between the two halves of a QI S-glass laminate. The results showed a pseudo-ductile behaviour with a linear elastic part and a desirable plateau for all the loading directions, however the pseudo-ductile strain decreases when increasing the off-axis angle. Comparing the 20° off-axis with the other cases, there was more active matrix cracking damage before fragmentation in the 20° off-axis plies and it failed earlier than the other samples. Acoustic emission (AE) results confirmed this, with more matrix cracking related AE signals in the 20° off-axis case compared to the other configurations.

Tidal Deformations of Hybrid Stars with Sharp Phase Transitions and Elastic Crusts

Abstract Gravitational wave astronomy is expected to provide independent constraints on neutron-star properties, such as their equation of state. This is possible with the measurements of binary components’ tidal deformability, which alter the point-particle gravitational waveforms of neutron-star binaries. Here, we provide a first study of the tidal deformability effects due to the elasticity/solidity of the crust (hadronic phase) in a hybrid neutron star, as well as the influence of a quark-hadronic phase density jump on tidal deformations. We employ the framework of non-radial perturbations with zero frequency and study hadronic phases presenting elastic aspects when perturbed (with the shear modulus approximately 1% of the pressure). We find that the relative tidal deformation change in a hybrid star with a perfect-fluid quark phase and a hadronic phase presenting an elastic part is never larger than about 2%–4% (with respect to a perfect-fluid counterpart). These maximum changes occur when the elastic region of a hybrid star is larger than approximately 60% of the star’s radius, which may happen when its quark phase is small and the density jump is large enough, or even when a hybrid star has an elastic mixed phase. For other cases, tidal deformation changes due to an elastic crust are negligible ( ) and, therefore, unlikely to be measured even with third generation detectors. Thus, only when the size of the elastic hadronic region of a hybrid star is over half of its radius, could the effects of elasticity have a noticeable impact on tidal deformations.

Positional finite element formulation for two-dimensional analysis of elasto-plastic solids with contact applied to cold forming processes simulation

In this work, we develop a position-based finite element formulation for elasto-plastic solids under contact situation. The proposed positional formulation employs a total Lagrangian description and naturally considers geometric nonlinearities. The employed elasto-plastic model is derived from the dissipation inequality, using the thermodynamic conjugacy between the plastic strain rate and the so-called Mandel stress. The formulation is based on the Kroner–Lee decomposition, in which the deformation gradient is multiplicatively split into its elastic and plastic parts. We apply the backward Euler method to solve the plastic evolutions and von Mises yield criterion to define the elastic limit. The adopted kinematic hardening model is a finite strain generalization of the Armstrong–Frederick law, which uses the objective Jaumann derivative for the evolution equation and the concept of back stress tensor as an internal variable. For the elastic parcel of strains, we adopt a neo-Hookean constitutive law. With respect to the 2D application, plane strain and plane stress approximations are considered, where the latter is solved numerically by a local Newton–Raphson numerical procedure. Regarding the contact problem, a classical node-to-segment algorithm is applied, considering both frictionless and frictional cases, with the introduction of Lagrange multipliers in order to enforce contact constraints. Representative numerical examples are used to validate and show the possibilities of the proposed formulation in macroscale simulation of metal cold forming manufacturing processes.

Finite strain phase-field microelasticity theory for modeling microstructural evolution

Implementing the phase-field model at finite strains is usually considered unattainable through the spectral method, largely because of the nonlinearity in the transformation-induced microelasticity. Here we present a phase-field microelasticity (PFM) theory at finite strains with a representation in the reference configuration, allowing the spectral method to be readily incorporated. Following the spirit of Khachaturyan’s PFM theory at small strains, the elastic energy is formulated as a functional of microstructure (order parameters) solely, which should automatically satisfy the mechanical equilibrium. Thermodynamic consistency of the current theory under multiplicative decomposition of the total deformation gradient (into elastic and inelastic parts) and in conjunction with hyperelasticity and the time-dependent Ginzburg-Landau equation is shown rigorously. The new theory is first applied to the classical Eshelby’s inclusion problem, where shear-dilation coupling due to geometric nonlinearity is shown and a convergence study between small strain and finite strain theories is also carried out. The effects of geometric nonlinearity on the co-evolution of micromechanics and microstructure is further studied through modeling the growth of {101¯2}〈1¯011〉 deformation twins in magnesium. The simulation results suggest significant differences in terms of the shape of and the stress field around the deformation twin. In particular, the current finite strain PFM theory predicts a deviation of the twin boundary plane from the theoretical K1 plane, which is not captured in the small strain theory nor in the crystallographic theory. A parametric study further reveals that the observed deviation is caused by the tip effect of the finite-sized twin plate when the aspect ratio is relatively small. The symmetry of the stress field distribution around the twin tip is also found to be drastically different between the small strain and finite strain based phase-field modeling. The sharp twin tip observed in experiments is also shown to be likely related to the anisotropy in twin/matrix interface mobility.

On Analytic Solutions of Elastic Net Displacements around a Circular Tunnel

Displacements around a tunnel, occurring as a result of excavation, consist of the elastic and plastic parts. In this paper, we discuss the elastic part of displacements as a result of excavation, called net displacement. In general, the previous analytical solutions presented for determining the displacements around a circular tunnel in an elastic medium do not give the net displacements directly. The well-known Kirsch solution is the most widely used method for determining the induced stresses and net displacements around a circular opening in a biaxially-loaded plate of homogeneous, isotropic, continuous, linearly elastic material. However, the complete solution for obtaining the net displacements has not been presented or highlighted in the available literature. Using the linear elasticity, this paper reviews and presents three different analytical methods for determining the net displacements directly as well as induced stresses around a circular tunnel. The three solution methods are the Lame' method, airy stress function method, and complex variable method. The tunnel is assumed to be situated in an elastic, continuum, and isotropic medium in the plane strain condition. The solutions are presented for both the hydrostatic and non-hydrostatic in situ stresses in the 2D biaxial loading condition along with an internal pressure. Loading and unloading in tunneling occurring as a result of excavation and stress differences between the induced and initial ones are considered to evaluate the net displacements directly. Finally, some examples are given to demonstrate the complete solution and show the difference between the net elastic displacements as a result of excavation and total elastic displacements that are not real.

A New Viscous Potential Function for Developing the Viscohyperelastic Constitutive Model for Bovine Liver Tissue: Continuum Formulation and Finite Element Implementation

The mechanical behavior of very soft tissues, such as liver, brain, kidney, etc. is assumed to be viscohyperelastic. The conventional approaches like quasi-linear viscoelastic (QLV) are limited to low strain rates and are unable to capture the short-term memory effects. In this paper, a new viscohyperelastic constitutive model is presented in which the strain energy function is decomposed into an elastic and a viscous part. The elastic part of the strain energy is assumed as a Mooney–Rivlin model, while a new viscous potential, as a function of strain and its rate, is proposed. Unconfined uniaxial compression tests are conducted up to 10% compression at two loading velocities of 1.3 and 10.56[Formula: see text](mm/min), to determine the material constants. A numerical simulation is also used to investigate the model-predicted material behavior. It is shown that the model is more sensitive to the hyperelastic parameters than the viscous parameters. This model is also able to predict the stress relaxation and hysteresis; however, an instantaneous relaxation is observed.

The generalized viscoelastic spring

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores in the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This paper begins with considerations compatible with large viscoelastic deformations and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed-form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress.

Equilibrium problem for elastic body with delaminated T-shape inclusion

We analyze an equilibrium problem for 2D elastic body with a T-shape thin inclusion in presence of damage. A part of the inclusion is elastic, and the other part is a rigid one. A delamination of the inclusion from the elastic body is assumed, thus forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The damage is characterized by a positive parameter. The paper provides an asymptotic analysis of the solutions as the damage parameter tends to infinity and to zero. A passage to infinity of a rigidity parameter of the elastic part of the inclusion is also analyzed. Junction conditions are determined at the connection point between the elastic and rigid parts of the inclusion. An existence theorem is proved for an inverse problem of finding displacement fields and the damage and rigidity parameters provided that a displacement of the tip point of the inclusion is known.

MAP123-EP: A mechanistic-based data-driven approach for numerical elastoplastic analysis

In this paper, a mechanistic-based data-driven approach, MAP123-EP, is proposed for numerical analysis of elastoplastic materials. In this method, stress-update is driven by a set of one-dimensional stress–strain data generated by numerical or physical experiments under uniaxial loading. Numerical results indicate that combined with the classical strain-driven scheme, the proposed method can predict the mechanical response of isotropic elastoplastic materials (characterized by J2 plasticity model with isotropic/kinematic hardening and associated Drucker–Prager model) accurately without resorting to the typical ingredients of classical model-based plasticity, such as decomposing the total strain into elastic and plastic parts, as well as identifying explicit functional expressions of yielding surface and hardening curve. This mechanistic-based data-driven approach has the potential of opening up a new avenue for numerical analysis of problems where complex material behaviors cannot be described in explicit function/functional forms. The applicability and limitation of the proposed approach are also discussed.

Transition from purely elastic to viscoelastic behavior of silica optical fibers at high temperatures characterized using regenerated Bragg gratings.

In this study, the response of regenerated fiber Bragg gratings (RFGBs) to axial forces was investigated in a temperature range from room temperature to 900 °C. For the first time, the transition from pure elastic to viscoelastic behavior around 700 °C of a standard SMF28 optical fiber was measured with an inscribed RFBG. An elastic model with linear temperature dependencies of Young's modulus and Poisson's ratio was established, and showed good agreement with the measurements up to temperatures of ∼500 °C. In the temperature range up to 900 °C, the RFBG response could be well described with a simple, single-material approach and a Burgers model that consists of a Kelvin and a Maxwell part. Based on the elastic parameter of the Maxwell part, the temperature-dependent force sensitivity of the RFBG was determined, and it showed a linear decrease in the range from room temperature to ∼500 °C, constant values in the range between ∼500 °C and ∼600 °C, and a strong increase at higher temperatures. While fulfilling the condition to operate in the elastic domain of the silica fiber, the investigations demonstrate that RFBGs can be used as force sensors up to temperatures of ∼600 °C - the range in which temperature-dependent force sensitivities have to be considered. The temperature-dependent parameters of the effective single-material model (elastic and viscoelastic part) are essential to describe the effective mechanical behavior of the optical fiber at high temperatures.

An analytical method for determining the non-enclosed elastoplastic interface of a circular hole

Based on the Mohr–Coulomb yield criterion, an analytical method is presented to determine the plastic zone in an infinite plate weakened by a circular hole and subjected to non-hydrostatic stresses at infinity. It is worth noting that this paper considers the more complicated case that the plastic zone cannot completely surround the hole, namely the elastoplastic interface is non-enclosed. Initially, the non-circular elastic zone in the physical plane is mapped onto the outer region of a unit circle in the image plane by the conformal transformation in the complex variable method. Thereby, determining the elastoplastic interface is equivalent to solving the mapping function coefficients. The nonlinear equations for solving the coefficients are established by considering both the stress continuity conditions along the elastoplastic interface and the stress boundary conditions along the elastic part of the hole. Naturally, the problem can be further transformed into an optimization problem, which is ultimately achieved by the differential-evolution algorithm; what is more, an analytical solution with high accuracy is obtained. Based on the programmed computing, the influences of various parameters on the shape and size of the plastic zone are given.

Micromechanical progressive damage analysis of inter- and intra-layer failures in fiber-reinforced composite laminates

We analyze progressive damage and failure of composite laminates by using a micromechanical bridging model and compare numerical and experimental results for three fracture problems, namely, the four-point bending, the simple tension, and the simple tension of a laminate with an open hole at its centroid. These problems involve fiber–matrix interface debonding, constituents’ damage, interlayer delamination, and localized damage due to stress concentration. Macroscopic constitutive equations of unidirectional lamina, derived from those of the fiber and the matrix by using the bridging model with the fiber material assumed to be linearly elastic and the matrix to be elasto-plastic obeying the Drucker–Prager yield criterion, are employed. Strains in each constituent of the composite are assumed to be infinitesimal for the additive decomposition of strains into elastic and plastic parts to be valid, and the incremental plasticity theory is used. Stresses in the two constituents are found from their values in the homogenized material by using a dehomogenization technique. The intra-layer damage is assumed to initiate at a material point when the failure criterion for either the fiber or the matrix is satisfied. Young’s modulus of a constituent is degraded by following a Weibull distribution. A finite element is deleted when an energy-based failure criterion is satisfied in it, and the analysis is continued till the structure fails. The delamination between adjacent plies is simulated by including a thin resin layer at the interface and studying failure initiation and propagation in it. The computed reaction force versus the displacement curves and the failure patterns in the three problems are found to agree with the corresponding experimental data.

Large deformation plasticity

The theory of plasticity as a special field of continuum mechanics deals with the irreversible, i.e. permanent, deformation of solids. Under the action of given loads or deformations, the state of the stresses and strains or the strain rates in these bodies is described. In this way, it complements the theory of elasticity for the reversible behavior of solids. In practice, it has been observed that many materials behave elastically up to a certain load (yield point), beyond that load, however, increasingly plastic or liquid-like. The combination of these two material properties is known as elastoplasticity. The classical elastoplastic material behavior is assumed to be time-independent or rate-independent. In contrast, we call a time- or rate-dependent behavior visco-elastoplastic and visco-plastic—if the elastic part of the deformation is neglected. In plasticity theory, because of the given loads the states of the state variables stress, strain and temperature as well as their changes are described. For this purpose, the observed phenomena are introduced and put into mathematical relationships. The constitutive relations describing the specific material behavior are finally embedded in the fundamental relations of continuum theory and physics. Historically, the theory of plasticity was introduced in order to better estimate the strength of constructions. An analysis based purely on elastic codes is not in a position to do this, and can occasionally even lead to incorrect interpretations. On the other hand, the entire field of forming techniques requires a theory for the description of plastic behavior. Starting from the classical description of plastic behavior with small deformations, the present review is intended to provide an insight into the state of the art when taking into account finite deformations.

An intergranular strain concept for material models formulated as rate equations.

SummaryThe intergranular strain concept was originally developed to capture the small‐strain behaviour of the soil with hypoplastic models. A change of the deformation direction leads to an increase of the material stiffness. To obtain elastic behaviour for smallstrains, only the elastic part of the material stiffness matrix is used. Two different approaches for an application of this concept to nonhypoplastic models are presented in this article. These approaches differ in the determination of the elastic stress response, which is used for reversible deformations. The first approach determines an elastic response from the original material model, and the second one uses an additional elastic model. Both approaches are applied on barodesy. The simulations are compared with experimental results and with simulations using hypoplastic models with the original intergranular strain concept.