Elastoplastic Body Research Articles

Development of plastic zones in a body with rectangular slot under the conditions of antiplane deformation

We study the quasistatic development of plastic zones in a perfectly elastoplastic body with rectangular slot caused by the action constant shear forces applied at infinity. The exact analytic solution of the problem is obtained, and it is shown that the plastic zones can be studied in the initial stage of development on the basis of the elastic solution of the problem. The shapes of the plastic zones are analyzed depending on the loads for the cases of cracklike, wide, and square slots.

On the direction of development of a thin fracture process zone at the tip of an interfacial crack between dissimilar media

The Wiener-Hopf method is used to study, under the conditions of plane strain, the direction of development of a thin fracture process zone at the tip of an interfacial crack in a piecewise homogeneous isotropic elastoplastic body. The zone is modeled by a straight line of tangential displacement discontinuity that emerges from the crack tip at an angle to the interface. The dependences of the zone length and the angle on the load and other parameters of the problem are investigated

An observation on the physical essence of elastoplastic deformations

AbstractOn account of the basic physical feature of elastoplastic deformations, namely, the occurrence of internal self‐equilibrated stress field upon removal of external loadings, a process of destressing an elastoplastic body with non‐uniform stress field would inevitably lead to loss of the line element concept in the resultant incompatible unstressed configuration. This would mean the loss of the kinematic ground or prerequisite for defining kinematic quantities in usual material continua. Then, an additional variable labelled as “elastic” or “plastic” via destressing could not be endowed with the necessary kinematic and physical content characterising a deformation quantity and, accordingly, might actually be a formal variable in mathematical sense. Thus follows an observation that the elastoplastic deformation would be an inherently inseparable physical entity. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the deformation of an elastoplastic body when placed between flat plates rotating about non-coincident axes

While several boundary value problems concerning the inelastic response of bodies have been studied analytically within the context of small strains, there are few studies that have been dedicated to such bodies undergoing large deformations despite the fact that numerous theories have been advanced to describe large inelastic deformations. Here, we discuss a solution to a boundary value problem (using an elastoplastic model for a polymeric material) wherein a layer is subject to a large deformation due to the rotation of two bounding planes of the layer. While the amount of rotation of the two planes is the same, the axes of rotation of the planes, while being parallel to one another, are non-coincident. The deformation under consideration amounts to a shear followed by a large rotation. It transpires that although the body is subject to a cyclic deformation, there is no unloading, i.e., the body, once it yields, continues to do so under deformation. The problem also highlights the shortcoming of a purely Eulerian approach to studying the inelastic response of bodies as the resulting permanent deformation and change of shape can be understood only in terms of a referential (or Lagrangian) approach. The solution also highlights the development of forces in a direction perpendicular to the plane of shear, namely the Poynting effect that is characteristic of the response of non-linear materials.

Two-Parameter Model of a Mode I Crack in an Elastoplastic Body under Plane-Strain conditions

The Dugdale crack model is generalized to the case of plane strain. The governing equations are set up to determine the stresses in the plastic zone. Numerical results from specific problems are analyzed and compared with those for plane stress state and other cases. A relationship between the crack model and KI-T theory is established in the case of small-scale yielding at the crack tip

2D phenomenological model for transversal deformations of superconducting cables

A theory of microplasticity for anisotropic elastoplastic body is used for phenomenological description and analysis of superconducting cables under transversal loading. It reflects the possibility to have hysteresis loops on the curve of loading and unloading on any level of loading including small ones. It reflects further the fact that these hysteresis loops don't depend on the strain rate. At last it reflects phenomenological anisotropy which is a characteristic property of the superconducting cable, because latter is produced of a large number of strands, oriented along the cable axis. The problem of identification of the model parameters is formulated and solved.

A quasi‐static boundary value problem in multi‐surface elastoplasticity: Part 1—Analysis

AbstractThe quasi‐static evolution of an elastoplastic body with a multi‐surface constitutive law of linear kinematic hardening type allows the modelling of curved stress–strain relations. It generalizes classical small‐strain elastoplasticity from one to various plastic phases. This paper presents the mathematical models and proves existence and uniqueness of the solution of the corresponding initial‐boundary value problem. The analysis involves an explicit estimate for the effective ellipticity constant. Copyright © 2004 John Wiley & Sons, Ltd.

Constitutive Model of High-Damping Rubber Materials

A mathematical model of high-damping rubber materials is developed. First the material experiments necessary for modeling are systematically conducted. Then, on the basis of the results of the material experiments, a constitutive model for rubber materials is proposed. The model is decomposed into two parts. The first part consists of an elastoplastic body with a strain-dependent isotropic hardening law and it represents the energy dissipation of the material, while the second part consists of a hyperelastic body with a damage model and it expresses the evolutional direction of the stress tensor. By comparing the experimental results with the simulation by the model, the model is found to well approximate the behaviors of high-damping rubber materials. Finally, a hybrid analysis method is proposed. In this method, the strain field of laminated rubber bearings measured by image processing is combined with the numerical analysis to confirm the applicability of the proposed model to the bearing. In addition, by this hybrid analysis method, the bulk modulus of rubber material is also computed.

3D rheological model of transversal isotropic elastoplastic body

AbstractA 3D theory of microplasticity for an anisotropic elastoplastic body is presented. This theory is proposed for the phenomenological description and analysis of deformation of a superconducting cable under the transversal cyclic loading. Obtained constitutive equations reflect the possibility to describe hysteresis loops during cyclic loading of the material. They reflect the fact that these hysteresis loops don't depend on the strain rate. Developed theory deals with phenomenological anisotropy of elastoplasticity which is a characteristic property of the superconducting cable. This cable consists of a large number of strands oriented in general along the cable axis.

State of Limiting Equilibrium of Cracked Bodies for Mixed Macromechanisms of Fracture

On the basis of the balance of works of plastic strains and fracture in a plastic zone formed near the crack front in an elastoplastic body, we establish and experimentally check analytic criteria of the limiting state and determine the direction of initial propagation of cracks under the conditions of complex stressed states in the case of possible realization of three macromechanisms (modes I + II + III) simultaneously. The proposed criteria are checked by using experimental procedures specially developed for the realization of the macromechanisms of fracture.

Development of Plastic Bands in the Process of Shear near the Tips of a Rectangular Slot

We study the development of rectilinear plastic bands from the tips of a rectangular slot of constant width in a perfectly elastoplastic body in shear. The plastic bands are formed along the bisectrices of the angles of the slot but, as the load increases, their orientation gradually changes and, finally, they become oriented along the axis of the slot. The slots of greater width are characterized by larger ranges of loads in which the bands propagate almost rectilinearly.

Concerning the fictitious continuum in damage mechanics

Consistent theories to describe damage processes are generally presented within the framework of effective stress and internal parameters. It is well known that damage is concerned with the progressive deterioration of elastic properties due to microscopic defects, such microvoids or microcracks. In the framework of Continuum Mechanics, damage is related to irreversible changes (on the microlevel) of small vicinities surrounding material points in the body. So a convenient definition of these small vicinities, named “representative material element”, will be recalled in Part 1, and application will be made to elastoplasticity in Part 2. In the subsequent parts, a fictitious suitable undamaged elastoplastic body accompanying the real damaged one is introduced in order to define the effective stress in the framework of large strains and its use in the construction of damaged elasticity law. Finally application is made to infinitesimal strains that concern most of the examples in literature. Due to limitation of place, plasticity coupled with damage is not considered in this paper.

The formation of a one-dimensional residual stress field in the neighbourhood of a cylindrical defect in the continuity of an elastoplastic medium

The one-dimensional finite deformations of the elastoplastic material of a thin-walled tube accompanying the loading of its external cylindrical surface are investigated. It is observed that, under conditions of total unloading when the external pressure is removed, the onset of a repeated plastic flow is possible, which changes the form of the distribution of the residual stresses considerably. The effect of the adaptability of an elastoplastic body with a cylindrical surface to repeated loading is pointed out.

Creep-rupture lifetime simulation of unidirectional metal matrix composites with and without time-dependent fiber breakage

Abstract A numerical simulation for predicting the axial creep-rupture lifetime of continuous fiber-reinforced metal matrix composites is proposed, based on the finite element method. The simulation model is composed of line elements representing the fibers and four-node isoparametric plane elements representing the matrix. While the fibers behave as an elastic body at all times, the matrix behaves as an elasto-plastic body at the loading process and an elasto-plastic creep body at the creep process. It is further assumed in the simulation that the fibers are fractured not only in stress criterion but time-dependently with random nature. Simulation results were compared with the creep-rupture lifetime data of a boron-aluminum composite with 10% fiber volume fraction experimentally obtained. The simulated creep-rupture lifetimes agreed well with the averages of the experimental data. The proposed simulation is further carried out to predict a possibility of creep-rupture for the composite without time-dependent fiber breakage. It is finally concluded that the creep-rupture of a boron-aluminum composite is closely related with the shear stress relaxation occurring in the matrix as well as time-dependent fiber breakage.

CONSTITUTIVE MODEL OF HIGH DAMPING RUBBER MATERIALS

A mathematical model of high-damping rubber materials is developed. First the material experiments necessary for modeling are systematically conducted. Then, on the basis of the results of the material experiments, a constitutive model for rubber materials is proposed. The model is decomposed into two parts. The first part consists of an elastoplastic body with a strain-dependent isotropic hardening law and it represents the energy dissipation of the material, while the second part consists of a hyperelastic body with a damage model and it expresses the evolutional direction of the stress tensor. By comparing the experimental results with the simulation by the model, the model is found to well approximate the behaviors of high-damping rubber materials. Finally, a hybrid analysis method is proposed. In this method, the strain field of laminated rubber bearings measured by image processing is combined with the numerical analysis to confirm the applicability of the proposed model to the bearing. In addition, by this hybrid analysis method, the bulk modulus of rubber material is also computed.

Analytical Solution of a Borehole Problem Using Strain Gradient Plasticity

An analytical solution is presented for the borehole problem of an elasto-plastic plane strain body containing a traction-free circular hole and subjected to uniform far field stress. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. This gradient plasticity theory introduces a higher-order spatial gradient of the effective plastic strain into the yield condition to account for the nonlocal interactions among material points, while leaving other relations in classical plasticity unaltered. The solution gives explicit expressions for the stress, strain, and displacement components. The hole radius enters these expressions not only in nondimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size effect in a quantitative manner. The classical plasticity-based solution of the borehole problem is obtained as a special case of the present solution. Numerical results for the plastic region radius and the stress concentration factor are provided to illustrate the application and significance of the newly derived solution.

Three‐dimensional stochastic finite element method for elasto‐plastic bodies

AbstractA new stochastic finite element method (SFEM) is formulated for three‐dimensional softening elasto‐plastic bodies with random material properties. The method is based on the Karhunen–Loeve and polynomial chaos expansions, and able to efficiently estimate complete probabilistic characteristics of the response, such as moments or PDFs. To reduce the computational complexity in the three‐dimensional setting, two alterations are made with respect to the two‐dimensional SFEM proposed earlier by the authors. First, a variability preserving modification of the Karhunen–Loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties. Second, an efficient algorithm for parallel processing is developed, with time consumption being the same order as for an ordinary FEM, rendering the proposed SFEM an effective alternative to Monte‐Carlo simulation. The applicability of the proposed method to stochastic analysis of strain localization is examined using Monte‐Carlo simulation. Then, it is applied to a fault formation problem which is a recent concern of earthquake engineering. Ground surface layers are modelled by a softening elasto‐plastic body, and the evolution of probabilistic characteristics of the rupture process is analysed in detail. Some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint. Copyright © 2001 John Wiley & Sons, Ltd.

Using comparison theorem to compare corotational stress rates in the model of perfect elastoplasticity

For the simple shear problem of a perfectly elastoplastic body, we convert the non-linear governing equations into a third order linear differential system, then into a second order linear differential system, and further into a Sturm–Liouville equation. Thus Sturm's comparison theorem can be employed and extended to compare the simple shear responses based on different objective corotational stress rates. It is proved that the rates of Jaumann, Green–Naghdi, Sowerby–Chu, Xiao–Bruhns–Meyers, and Lee–Mallett–Wertheimer render non-oscillatory stress responses, with the Jaumann equation as a Sturm majorant for the other four equations. For an objective corotational stress rate with the general plane spin a sufficient non-oscillation criterion is found to be that the plane spin must not exceed the shear strain rate.

Distortion of the Wave Profiles in an Elastoplastic Body upon Spalling

The distortion of wave profiles in measuring the spall strength of elastoplastic materials is analyzed. An expression for the velocity of an elastic compression wave that overtakes a plastic rarefaction wave is obtained. It is shown that, depending on the ratio between the stress gradients in the plastic rarefaction wave and the overtaking compression wave, the front velocity of the compressive wave varies in the limits between the velocities of the longitudinal perturbations and the perturbations of volume expansion or compression.

地表地震断層シミュレ-ションのための弾塑性確率有限要素法とその計算例

In order to simulate the formation process of surface earthquake faults, we develop a new stochastic finite element method for an elasto-plastic body with randomly varying material properties and apply it to the 3D Drucker-Prager model. The method is based on the Karhunen- Loeve expansion and Polynomial chaos expansion to efficiently estimate complete probabilistic characteristics of responses, such as average, variance, and a probability density function. Also, it is shown that the proposed stochastic finite element method is able to simulate complicated bifurcation phenomena associate with the fault formation process.