Rigid-perfectly Plastic Research Articles

Predicting the dynamic response of elastic-plastic beams by dimensional analysis

Elastic-plastic beams subjected to dynamic loading are widely found in engineering. However, due to the complex coupling of geometric and material nonlinearity, there is no complete analytical solution available. In this paper, based on energy conservation and dimensional analysis, a dominant dimensionless number, the loading intensity ξ, is proposed. Including both the effects of geometry, material and load, ξ could be used to predict the significant responses of an elastic-plastic beam under uniformly pressure loading, such as energy ratio, deflection ratio and deformation mechanism. With combination of loading intensity ξ and dimensionless stiffness β, the dimensionless maximum and final deflections of elastic-plastic beams under pressure loading can be predicted directly. Dimensionless numbers of beams with different cross-sections, or under different loading forms are also analyzed and confirmed.Then, elastic effect over dynamic response of elastic-plastic beams is analyzed. For final deflection, it is found ξ>5 is the applicability of theoretical solution based on rigid, perfectly plastic (R-PP) material simplification. However, R-PP results can't be used in initial plastic hinge location and subsequent plastic zone evolution, even for very large ξ.Using these dimensionless numbers, experimental or simulation data could be expressed in a normalized and comparable form, and the physical mechanisms will be more intuitive and clearer, which is valuable for scaled dynamic tests.

Mastering the art: Proficient finite element implementation and robust evaluation of a strain-hardening porous ductile material crack growth prediction model at finite strain

Accurate prediction of crack extension is crucial for ensuring the structural integrity of metal components exposed to diverse loads. While the Gurson model and its extensions are widely accepted for delineating ductile fracture stages, especially in porous materials with a rigid-perfectly plastic matrix, their efficacy diminishes in the presence of strain-hardening matrices. To address this limitation, our study introduces a robust numerical implementation and assessment of Perrin's model. In contrast to Gurson's approach, Perrin's model incorporates two distinct strain-hardening parameters derived from an intricate analysis of a strain-hardening hollow sphere subjected to axisymmetric loading.Moving beyond theoretical discussions, this paper actively engages in the rigorous evaluation of Perrin's model under various scenarios, encompassing isotropic, kinematic, and mixed isotropic-kinematic hardening conditions. The model's effectiveness is convincingly demonstrated through meticulous comparisons between numerical simulations and real-world experimental observations conducted on pre-cracked specimens.Facing challenges related to achieving global elastic-plastic convergence in large-scale simulations, our research introduces a sophisticated approach. Leveraging stiffness tangent moduli ensure the maintenance of quadratic convergence in global Newton method iterations. This not only enhances the reliability of our simulations but also underscores the practical applicability of our findings.The study also focuses on the necking and crack propagation of a cylindrical specimen through meticulous computational modeling. Our mesh captures intricate details and addresses stress gradients, revealing coplanar crack propagation contrary to prior beliefs. The confirmation of the cup-cone fracture effect challenges previous interpretations, emphasizing the critical role of void nucleation. This reinforces the credibility and applicability of our approach in advancing the understanding of material fracture behavior.In summary, this study significantly contributes to existing knowledge in fracture prediction models, offering a robust framework for predicting ductile fracture under substantial deformations. The presented findings pave the way for advancements in structural engineering, providing invaluable insights for optimizing the performance and resilience of metal structures in real-world applications.

Analytical study on the dynamic mechanical behaviours of foam-core sandwich plate under repeated impacts

The foam-core sandwich structures are probably subjected to repeated impact loadings in engineering applications, resulting in structural plastic deformation and accumulative damage. A valuable analytical approach is needed to predict the accumulative plastic deformation of foam-core sandwich structures under repeated impacts. In this paper, an analytical model of the dynamic responses of foam-core sandwich plates under repeated impacts is proposed using the circumscribing yield locus and the inscribing yield locus based on the rigid-perfectly plastic theory. By employing the principle of energy dissipation rate balance, the analytical expressions of structure permanent deformation and peak force are obtained from the iteration of initial conditions. In addition, polyvinyl chloride (PVC) foam has steady chemical properties and corrosion resistance, which can better adapt to the practical environment and meet the needs of structural protection. To validate the proposed analytical model, the repeated impact dynamic mechanical behaviours of PVC foam-core sandwich plates were experimentally and numerically studied. Results show that theoretical results such as the permanent deformation, peak force, and deformation modes of foam-core sandwich plates under repeated impacts are in good agreement with experimental and numerical results. With the increase in impact numbers, the global bending deformation and local indentation deformation of facesheets increase, the PVC foam core layer is constantly being compressed, and the bottom facesheet plays an increasingly important part in the repeated impact energy absorption with a maximum value of 29.20 %.

Upper Bound Limit Analysis of 2D Structures Using Lagrange Extraction-Based Isogeometric Analysis

This work presents a new approach that uses Lagrange extraction-based isogeometric analysis and Second Order Cone Programming (SOCP) to determine the limit load factor in two-dimensional problems. This methodology makes use of an upper-bound limit approach and a rigid-perfectly plastic material model. Lagrange extraction is used to link a C0 nodal basis to a smooth B-spline basis. This enables the use of straightforward methods for isogeometric analysis in conventional finite element systems. A SOCP problem is created from the limit analysis problem, which may then be solved using Mosek optimization software. The numerical results indicate the correctness and effectiveness of the current technique by contrasting it with other approaches mentioned in the literature.

Elastic effect on the final deflection of rigid-plastic square plates under pulse loading

Rigid-perfectly plastic (R-PP) models are widely employed in the study of structural impact dynamics. However, due to the complexity of dynamic elastic-plastic behavior and the limitations of numerical simulation, it is of great significance to evaluate the elastic effect on the structural response to pulse loading. This paper examines the validity of the R-PP theoretical solutions on large deflection of pulse-loaded square plates. First, by exploring the characteristic of “plastic membrane” and considering the effect of membrane forces and bending moments on energy dissipation, an energy ratio R and a characteristic time ratio are precisely defined. Then based on the R-PP complete solutions recently developed for square plates under rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse between the R-PP theoretical predictions and the elastic-perfectly plastic (E-PP) simulation results are evaluated and elaborated. It is revealed that the discrepancy in final deflection is dependent on the dimensionless stiffness ζplate and the dimensionless pulse magnitude λ. Moreover, the discrepancies in final deflection are compared between square plates and beams via a thorough analysis of key parameters, indicating that most characteristics of the elastic effect are similar for both cases, i.e., the R-PP solutions for both square plates and beams can reliably provide theoretical predictions for engineering applications if energy ratios R>5, with no limitation on the pulse duration. With the same geometric and material parameters, however, the energy ratio R of square plates is about twice that of beams, the discrepancy in final deflection of plates is larger than that of beams, but the absolute value of the discrepancy in final deflection for the clamped square plates remains in the order of 1/(1.5R) and can be bounded by 1/[4λ(λ-1)].

Confined buckle propagation of offshore pipelines under external pressure

Offshore pipelines can be protected against collapse and buckling propagation by installing buckle arrestors or restraint cylinders. When the external pressure is high enough, the buckling would penetrate the arrestors or restraint cylinders and propagate along the line inside these protective structures, known as confined buckle propagation. A two-dimensional theoretical model has been proposed to estimate the confined propagation pressure, considering the rigid-perfectly plastic model and the Ramberg-Osgood model, respectively. Subsequently, the hyperbaric chamber test on SS304 confined tubes was conducted under external pressure. The annulus space of test specimens between the internal pipe and the external constraint cylinder was full of high-strength cement mortar and epoxy resin, respectively. Numerical models have been developed to duplicate the local collapse and the confined buckle propagation scenarios, and reasonable consistencies are observed between the tests and the numerical simulations. After that, broad parametric analyses of the confined propagation pressure are implemented to examine the influence of geometry and material property. As a result, an empirical expression is proposed to predict the confined propagation pressure.

On the yield criterion of porous materials by the homogenization approach and Steigmann–Ogden surface model

In this work, we investigate the yield criterion of nanoporous materials by using homogenization approach and Steigmann–Ogden surface model. The representative volume element is proposed as an infinite matrix containing a tiny nanovoid. The matrix is incompressible, rigid-perfectly plastic, von Mises materials and nanovoids are dilute and equal in size. First, the constitutive of microscopic stress and microscopic strain rate is established based on the flow criterion. Secondly, according to the Hill’s lemma, the relationship between the macroscopic equivalent modulus and the microscopic equivalent modulus is established by homogenization approach. Thirdly, the macroscopic equivalent modulus containing the Steigmann–Ogden surface model including surface parameters, porosity and nanovoid radius is derived from the trial microscopic velocity field. Finally, an implicit macroscopic yield criterion for nanoporous materials is developed. For surface modulus, nanovoids radius and porosity studies are developed through extensive numerical experiments. The research results in this paper have reference significance for the design and manufacture of nanoporous materials.

Structural response analysis of free rigid-perfectly plastic warhead under three typical transverse loads

Based on the rigid-perfectly plastic assumption, the structural response of a free warhead under the application of concentrated force, evenly distributed force, and ramping-up force was analysed, taking the geometrical features into account. By the sectional shear force and bending moment distribution, the critical section was determined through a yield criterion based on the fully plastic bending moment. It shows that when the load is applied at the tip of the warhead, singularity occurs in the yield function at the tip, but vanishes when the applying (or starting) point of the load some distance away. The influence of the distance of the applying (or starting) point to the tip was also discussed, and was found to be important in determining the location of the critical section. As the applying (or starting) point moves away from the tip, the critical section appears near the tip first, then jumps to the cylinder part of the warhead, and when far enough, jumps back to the head of the warhead in the concentrated force case, while in the distributed force cases the critical section appears first at the cylinder part, and then jumps to the head. The critical section jumps to the head much earlier and more localized around the connecting point of the head and the cylinder in the ramping-up case than in the evenly distributed case.

Elastic effects on the dynamic plastic deflection of pulse-loaded beams

Beams under intense dynamic pulse loading may experience large elastic-plastic deformation. Despite numerous studies have been conducted based on the rigid-perfectly plastic idealization of material, it is of great interest in both theoretical modeling and practical application to characterize and evaluate the effect of elastic deformation on the dynamic behavior of structures. This paper is aimed to examine the range of applicability of rigid-plastic theoretical solutions on large deflection of beams in response to dynamic pulse loading. The energy ratio R and the characteristic time ratio are first precisely defined by referring to the characteristics of a “plastic string”, which represents the ultimate status of beams under very large deformation. Based on the recently published complete solutions on the dynamic response of fully-clamped and simply-supported rigid-plastic beams subjected to uniformly distributed rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse of the rigid-plastic theoretical predictions from the elastic-plastic simulation results are evaluated and analyzed. It is revealed that if the energy ratio R>5 (with no limitation to the pulse duration), the rigid-plastic solutions can provide reliable predictions on final deflection and saturated impulse for the usage in engineering design. Then, the effects of the key dimensionless parameters related to the energy ratio R (e.g., the structural and material parameters as well as the loading one) on the discrepancies in final deflection and in saturated impulse are thoroughly examined. Finally, a unified dimensionless stiffness is identified to characterize the combined effect of structural and material parameters. For the cases of ζ≤1.581, which represent most of the practically used beams, 1/(2R) and 1/(3R) could provide upper bounds to the absolute value of the discrepancy in final deflection for fully-clamped and simply-supported beams, respectively.

Low Temperature Effect on the Mechanical Properties of EH36 with Strain Rates

With the expansion of the Arctic route, the safety of ship crossing the area in light of the low temperature and ice has become of focus, especially with regards to the ship’s structure. The mechanical properties of the material making up the ship’s structure may not be suitable for the Arctic environment. A series of quasi-static and dynamic tests were performed to investigate the behaviour of EH36 steel, which is used to build Arctic ships, at temperatures ranging from 20 °C to −60 °C. The yield and ultimate tensile stress increased more than 10% as the temperature decreased from 20 °C to −60 °C, whereas the toughness decreased as the temperature decreased. A formula was derived to illustrate the relationship between the temperature reduction and the yield strength by fitting the experimental data. Four common constitutive rigid-perfectly plastic, elastic-perfectly plastic, bilinear elastic-plastic, and multi-linear elastic plastic models were fitted to simulate the hull structure under static loading and low temperature. Additionally, the strain rate effect of EH36 steel at low temperatures was illustrated by quasi-static and high-speed impact tests. A constitutive model including the low temperature and strain rate was introduced based on a modified Cowper-Symonds model, in which the coefficients of the constitutive model are fitted by the test results. It is improved by an iterative numerical method used to obtain more accurate coefficients using a series of numerical analyses. Detailed finite element simulations of the experiment conditions revealed that the constitutive model accurately predicts the dynamic response at low temperatures.

Analytical study on the dynamic response of foam-core sandwich plate under wedge impact

In this paper, the dynamic responses of sandwich plate with foam core subjected to wedge impact are theoretically investigated. An analytical model of fully clamped foam-core sandwich plate under low-velocity wedge impact is developed based on the approximate yield criteria and rigid-perfectly plastic theory. Besides, the horizontal impact experiments and numerical simulations of polyvinyl chloride (PVC) foam-core sandwich plates were conducted to verify the accuracy of analytical prediction model. Results show that the analytical prediction results of foam-core sandwich plate under wedge impact such as the permanent deformations and impact force as well as deformation distribution are consistent with the experimental and numerical results. Furthermore, the geometrical parameters such as wedge width and aspect ratio of sandwich plate as well as foam core thickness on the dynamic responses are theoretically performed. Results showed that the deformation reaches to maximum with aspect ratio of 1and the permanent deformation declines with the increase of wedge width and foam core thickness.

Seismic stability of a circular tunnel in cohesive-frictional soils using a stable node-based smoothed finite element method

This paper presents the effect of horizontal and vertical earthquake force on the stability of a single circular tunnel in cohesive-frictional soils using a stable node-based smoothed finite element method (SNS-FEM). In this study, seismic forces are computed as horizontal and vertical pseudo-static body forces arising on the soil and additional inertial forces associated with the uniform surcharge applied to the ground surface. In the upper bound limit analysis based on SNS-FEM, the soil behaviour is described as rigid-perfectly plastic materials, and plasticity deformation obeys the associated flow rule following the Mohr-Coulomb failure criterion. Firstly, the numerical results were checked against other numerical solutions in the literature. The present results agree with prior contributions, proving that the proposed approach can give efficient and reliable solutions to the stability number. Secondly, the variations of the seismic stability number with changes in the horizontal earthquake acceleration coefficient were intensively investigated for different values of soil properties, internal friction angleand the depth-to-diameter ratio of the tunnel. It is shown that the seismic stability numbers of circular tunnels reduce remarkably with the increase of horizontal seismic coefficientand the soil weight. Thirdly, the seismic stability numbers were summarised in design charts for practical use in geotechnical engineering.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.

In-plane pipe whip: Post-failure dynamic response

Pressurised pipes are ubiquitous in chemical, nuclear and power plants. A sudden failure and release of high-speed fluid cause large inelastic displacements characterised by a whipping-type motion, which can ultimately hinder the surrounding structural and functional systems. In this paper, a beam user element has been developed, implemented and applied to analyse the in-plane flexural dynamic response of pipe whip. The two-dimensional Euler–Bernoulli beam element is based on the corotational kinematic formulation and elastoplastic constitutive models that include metal plasticity and moment–curvature relationships obtained from numerical bending tests, which highlighted the existence of two new dimensionless groups that govern the flexural response of slender pipes and enable the creation of moment–curvature master curves for thick and thin pipes. The corotational beam element formulation is compared against an analytical rigid-perfectly plastic model, numerical simulations employing shell elements and available experimental results, showing very good accuracy in the prediction of the inelastic deformation response of pipe whip and its hazardous area of influence. Furthermore, parametric studies are performed to investigate the effect of load intensity, cross-sectional geometry and concentrated tip mass on the post-failure deformation modes, plastic hinge formation and extension of the hazard zone. The presented results represent valid tools to assess the safety of industrial piping systems undergoing failure, and to optimally design pipe whip restraints.

Mathematical modeling the rigid-plastic yielding behavior of fibrous flatly-reinforced composites of anisotropic materials at 2D stress state

Structural model of multi-directed flatly-reinforced hybrid composites allowing one to determine the yield loci for the composition with account plane stress state in all components is developed. The fiber and matrix materials are homogeneous and anisotropic. Their mechanical behavior is described by associated flow law for rigid-perfectly plastic body with quadratic Mises – Hill yield criterion. As examples, two variants of symmetrical reinforcement for compositions consisting of orthotropic materials are considered under condition of equal reinforcement consumption. The influence of anisotropy characteristics of components and reinforcement parameters on shape and dimensions of the yield loci of the compositions is investigated.

A novel device to improve robustness of end plate beam–column connections: Analytical model development

A novel device to increase rotational capacity of steel end plate beam–column connections is proposed. By inserting a steel sleeve with a designated length, thickness and wall curvature between the end plate and the washer, the load path between the end plate and the bolts can be interrupted, promoting a more ductile response. For a specific sleeve material and dimensions, the capacity of sleeve is controlled by the amplitude value. The sleeve develops severe plastic deformations and is crushed between the end plate and the washer at the plastic amplitude. This study proposes an analytical solution to predict the plastic amplitude for the sleeve with a circular wave form. The sleeve is mathematically represented using shell of revolution theories under axisymmetric load. To simplify the analysis, the sleeve material is analysed using the two-moment limited interaction yield condition and follows an idealised rigid-perfectly plastic material response. The comparison between the proposed analytical solution with validated FE models show that the plastic amplitude is accurately captured. However, the predicted rotational capacity is very conservative.

Load-bearing capacity of rigid-plastic reinforced shallow shells and plates

The extreme problem of finding upper (kinematic) limit load of rigid-plastic fiber-reinforced spherical shallow shells and plates, which are circular in plan, is formulated and solved. The structures are clamped and have rigid insert. Two-layer shell model is used. The materials of components are rigid-perfectly plastic and asymmetric in tension and compression. The binder is isotropic or orthotropic. The structural model of composition takes into account two-dimensional stress state in the components. The piecewise-linear yield curves are constructed for radial-circumferential and spiral reinforcement. Comparisons results with other studies are given and influence of reinforcement parameters on limit load is investigated.

On the strength of nanoporous materials with the account of surface effects

The increasing applications of nanoporous materials in engineering structures call for robust strength criteria that are able to model the size effects of multiscale voids inherent to these materials. Many literature works have devoted to this line of research by either employing the classical homogenization approach or extending Gurson’s model to incorporate the nanovoid surface mechanics. However, few studies are able to simultaneously account for the multiscale effects of both nanovoid and microvoid porosities. This work presents a two-level hierarchical strength theory for nanoporous materials accommodating the effects of nanovoid surface and multiscale porosities. No surface effects exist for microvoids because of their sufficiently large length scale. The lower level representative volume element (RVE) is modeled as a homogeneous, incompressible and rigid-perfectly plastic matrix embedded with multiple nanovoids of similar size. The lower level RVE is then treated as a material point in the upper level RVE made by a hollow sphere. By the coupling of homogenization method for the lower level RVE and the limit analysis of Gurson type at the upper level, an implicit closed-form macroscopic strength criterion is derived. Parametric studies about the significance of nanovoid surface properties, nanovoid size, nanovoid porosity and microvoid porosity on yield loci reveal that the proposed two-level hierarchical model is a substantial reinforcement to the conventional single level strength criteria available in the open literature.

Theoretical, emulation and experimental analysis on auxetic re-entrant octagonal honeycombs and its applications on pedestrian protection of engine hood

In this article, a novel re-entrant octagonal honeycomb (ROH) with negative Poisson's ratio (NPR) configuration is proposed owing to its superior energy absorption and lightweight design performances. Regarding the effects of adjacent unit cells, a mechanics-based homogenization approach is put forward and employed to derive static mechanical properties including elastic modulus and Poisson's ratio of two-dimensional (2D) and three-dimensional (3D) ROHs under axial loading. Based on above, the predominant compression failure mode is explored. Then, the static mechanical properties and finite element model of ROH are validated by experiment and demonstrate high accuracy. Subsequently, to obtain dynamic responses of the ROH under dynamic impact loading, the relations among dynamic plateau stress and quasi-static plateau stress, initial density, dense strain and impact velocities are established under the conditions of mass conservation and momentum theorem based on one-dimensional shock wave theory and ideal rigid-perfectly plastic material model. Besides, dynamic energy absorption models of the ROH under dynamic loading are acquired in consideration of the effects of impact velocities on dense strain, the results show that theoretical models agree quite well with simulation results. On this basis, the application of ROH on automotive energy absorber is studied with the purpose of acquiring superior energy absorption and satisfying lightweight requirements simultaneously. Based on the decoupling thought of stiffness and pedestrian protection, a new design method of carbon fiber reinforced plastics (CFRP) hood with ROH is put forward in consideration of energy absorption and lightweight requirements. Of this, ROH is regarded as sandwich structure to meet the demands of energy absorption, CFRP plates are treated as outer and inner skins to satisfy the needs of stiffness. Firstly, multi-objective optimization models are set up under the constraints of reasonable structural parameters and negative Poisson’s ratio. Then, the NSGA-Ⅱalgorithm is adopted to achieve the optimal Pareto fronts with the targets for maximum unit mass energy absorption and minimum density. Afterwards, the optimization on thicknesses of CFRP layers is carried out to achieve minimum mass under the constraint conditions that the static stiffness of CFRP hood is larger than that of steel hood. In addition, the layering sequence of inner and lower skins are also optimized. Finally, improved overall stiffness, optimized HIC values and similar lightweight space of CFRP hood with ROH are achieved in comparisons with CFRP hood without ROH. Compared with steel hood, the pedestrian protection performance of CFRP hood with ROH is greatly enhanced, 1st free mode increases by 33.9%, and torsional stiffness and bending stiffness increase by 23.1% and 30.8%, respectively. While the weight decreases by 40.1%. The results indicate that the CFRP hood with ROH presents outstanding advantages of pedestrian protection and lightweight design in terms of HIC and stiffness performance.

A continuum level-set model of fracture

This work is devoted to the modeling of brittle and ductile fracture under the use of the level-set method. Within the proposed model a level-set function is taken as a smooth function that represents brittle damage in an implicit manner, that is the zero level-set of the continuous function coincides with the boundaries of the damage. Under the utilization of a regularization parameter that can be interpreted as a material’s internal length, crack faces of zero width are replaced by interfaces of finite width scaled by the length parameter. An energy functional is adopted and after following a variational field approach the equations of damage nucleation and propagation are revealed. Simple numerical examples indicate that the proposed level-set model is in good agreement with other successful models for brittle fracture. Moreover, ductile fracture is successfully captured by introducing an additional level set function so as to describe three distinct phases within the body: the damaged area, the rigid-perfectly plastic region and the elastic region. A simple example of damage propagation provides the behavior of a ductile material.