Propagation Of Plastic Zone Research Articles

Non-linear behavior of functionally graded elastoplastic beam under torsion

The torsional behavior of beams graded in one and two directions under large displacements and angular deformations was analyzed using the power law and sinusoidal functions. Functionally graded material is elastoplastic, consisting of ceramic and metal. A nonlinear finite element method with isoparametric hexahedral elements was used. The finite element formulation was made using the updated Lagrangian formulation based on the virtual displacement principle. An iterative solution using Newton-Raphson and updated Newton-Raphson methods was used to solve the nonlinear equation system. The propagation of the plastic zone was calculated based on the flow theory of plasticity. Elastoplastic behavior and effective material properties were determined according to the TTO model. Numerical investigations have shown that functionally graded beams behave quite differently from homogeneous beams under torsion. Yielding of the material starts at the outer boundaries of the section of the homogeneous beams, and the plastic zone propagates symmetrically. On the other hand, yielding and propagation of plastic zones tend to shift to regions with more ceramic volume with higher effective Young modulus in functionally graded beams. Unlike homogeneous beams, the largest shear stresses can occur within the section rather than at the outer boundaries of the section. In beams graded from ceramic to metal using the power law, the section moves along the transverse direction in addition to the rotation. This transverse displacement occurs in the grading direction, and its magnitude is about 3% of the thickness at 12.5° rotation angle. Also, the shear stresses are not zero in the section's midpoint. The effects of material distribution on displacements, stresses, and plastic zone propagation were examined, and essential points were reported.

Failure mechanism and divisional differentiated control of surrounding rock in mining roadway under remaining coal pillar in close‐distance coal seam

AbstractRegarding the issue of intense mining pressure appearing in the underlying gateway below the remaining coal pillar in the close‐distance coal seam (the remaining coal pillar is perpendicular to the underlying section coal pillar), 401 working face is used as the engineering background. Field measurements, laboratory experiments, numerical simulations, and engineering verification techniques are used to study the abutment pressure's evolution properties and the plastic zone's propagation laws before and after the underlying coal seam roadway experienced the mining impact. The conclusions are as follows: ① The maximum plastic area on the two sides and the roof of the roadway underlying the gob are up to 2 and 1.5 m, whereas the maximum plastic area on the two sides and the roof of the roadway underlying the remaining coal pillar are up to 5 and 4.5 m, respectively. Moreover, the plastic area extends along the two sides, and the section coal pillar is completely broken when the working face is mined below the remaining coal pillar. ② The stress increase coefficient K in the overlap area of the remaining coal pillar and the underlying section coal pillar reaches 3.4 when the mining face penetrates the underlying remaining coal pillar and the advance abutment pressure is overlaid with the concentrated stress of the coal pillar. ③ When the underlying working face is mined to 4, −2, −8, and −14 m away from the remaining coal pillar, the damage range of the roadway 5–10 m ahead increases in turn. At the same time, the maximum plastic area of the roof passes through the plastic area of the upper coal seam floor. Therefore, the underlying and transition areas on both sides of the remaining coal pillar are divided into Area I (15 m) → Area II (the most complicated area to control under the remaining coal pillar, 20 m) → Area III (25 m) according to the width. Furthermore, the divisional differentiated combined control technology of channel steel truss anchor cable with joint double‐way locking control function of roof and coal pillar in Areas I and III, while channel steel truss anchor cable with joint double‐way locking control function of roof and side + high resistance integral door‐type support is proposed in Area II. Field engineering practice shows that the deformation of the roadway surrounding rock can be controlled within 210 mm after adopting the above divisional combined control technology. Finally, the mining operation can safely and efficiently pass through the remaining coal pillar. The research results have important reference values for surrounding rock control of mining roadways in the overlapping area of similar “+”‐type cross‐working face.

Criterion for elastoplastic failure of bimetallic plate with edge crack of transverse shear at the boundary of the materials joint

The initiation of a transverse shear edge crack in elastoplastic materials with the limiting strain is considered. The crack propagation criterion is formulated using a modified Leonov-Panasyuk-Dugdale model using the additional parameter - the width of the plasticity zone. The coupled quasi-brittle fracture criterion for mode II cracks in an elastoplastic material is formulated under conditions of small-scale yield in the presence of a singular feature of the stress field in the vicinity of the crack tip. The coupled fracture criterion includes the deformation criterion, which is formulated at the crack tip, as well as the force criterion, which is formulated at the model crack tip. The lengths of the initial and model cracks differ by the length of the prefracture zone. The sequential analysis of the possibility of applying the proposed fracture criterion in determining the critical loads for solids containing edge cracks of transverse shear at the interface of different media is performed. Quasi-brittle fracture diagrams are constructed for a composite plate with an edge crack under plane strain and plane stress conditions. he analysis of the parameters included in the proposed model of quasi-brittle fracture is carried out. The model parameters are proposed to be selected by approximation of simple shear diagram and critical stress intensity factor. The critical loads were found numerically for the quasi-ductile and ductile fracture types. The finite element method is used to solve the problem of drawing out a reinforcing layer from a metal composite under quasi-static loading. The process of propagation of plastic zones in the vicinity of the crack tip is described consistently. It is shown that the shapes of the constructed plastic zones differ significantly from the well-known classical concepts.

Simulation of fracture of elastoplastic materials in mode III: from brittle to ductile

The initiation of a longitudinal shear crack (mode III fracture) in elastoplastic materials with the limiting strain is considered. The crack propagation criterion is formulated using a modified Leonov–Panasyuk–Dugdale model using an additional parameter—the width of the plasticity zone (the width of the pre-fracture zone). Conditions of a small-scale yielding in the presence of a stress field singularity in the vicinity of the crack tip are considered. A two-parameter criterion for quasi-brittle fracture of an elastoplastic material is formulated for mode III cracks. The proposed fracture criterion includes the deformation-based criterion, which is formulated at the crack tip, as well as the force-based criterion, formulated at the tip of a model crack. The lengths of the initial and model cracks differ by the length of the pre-fracture zone. The application of the proposed strength criterion to the determination of fracture loads for bodies containing longitudinal shear cracks is demonstrated. The diagrams of quasi-brittle fracture are constructed for a strip of finite width with an edge crack in case of the out-of-plane deformation. It is proposed to determine the model parameters using an idealized simple shear diagram and the critical stress intensity factor. For quasi-ductile and ductile types of fracture, the limiting loads are determined numerically. The propagation of plastic zones in the vicinity of the crack tip under quasistatic loading is successively described by the nonlinear finite element method. It is shown that the shapes of the numerically simulated plastic zones differ significantly from the known classical concepts.

Nonlinear bending of elastoplastic functionally graded ceramic-metal beams subjected to nonuniform distributed loads

Nonlinear bending of elastoplastic functionally graded (FG) ceramic-metal beams subjected to nonuniform distributed loads is investigated by the finite element method. A bilinear stress-strain relation with isotropic hardening is assumed for elastoplastic behavior of metal, and the elastoplastic properties of the FG ceramic-metal material are evaluated by using Tamura–Tomota–Ozawa (TTO) model. Based on Euler–Bernoulli beam theory, a nonlinear finite element formulation, taking the effect of plastic deformation into account, is derived and used in the investigation. The formulation employing nonlinear von Kámán strain-displacement relationship is derived by using the physical neutral surface as reference plane. An incremental-iterative procedure based on Newton–Raphson method, in which the plastic equation is solved at Gauss points for updating the stress and evaluating the element formulation, is employed to solve nonlinear equilibrium equations. The elastoplastic behavior is illustrated for a FG beam composed of TiB and Ti. The numerical results show that yielding in the FG beam occurs at the layer near the ceramic surface earlier than it does at the layer near the metal surface. The effect of the material distribution, plastic deformation on the nonlinear behavior of the beam with various end conditions is investigated in detail. The formation and propagation of plastic zone inside the beam during the loading process is also examined and highlighted.

Elasto-plastic analysis of functionally graded metal-ceramic beams under mechanical loading

The elasto-plastic analysis of functionally graded (FG) metal-ceramic beams under mechanical loading by using the finite element method is presented. A bilinear stress-strain relation with isotropic hardening is assumed for elasto-plastic behaviour of metal, and the effective elasto-plastic properties of the functionally graded material are evaluated by using Tamura-Tomota-Ozawa (TTO) model. A nonlinear beam element based on the classical beam theory is formulated and employed in the analysis. The element employed nonlinear von K\'am\'an strain-displacement relationship is derived by using the neutral surface as reference plane. The layer beam approach, in which the plastic rate equation is solved at Gauss points, is adopted in updating the stress and evaluating the element nodal force vector and tangent stiffness matrix. Numerical examples are given to show the accuracy of the derived formulation and to illustrate the effect of the material distribution and plastic deformation on the behavior of the beams. The formation and propagation of plastic zone during the loading process is also examined and highlighted.

Mechanism of Subordinate Peak Skewing of FBG Sensor during Cracks Propagation Monitoring on Aluminum Alloy Structure

This study investigates the variety of the spectra features of fiber Bragg grating (FBG) around the crack tip during fatigue crack propagation. The study results reveal that the turning of the subordinate peak is significantly associated with crack lengths and corresponds to strain gradient along the FBG. Meanwhile, the strain distribution sensed by the FBG changes with the sensing section of the grating. FBG sensors could observe the monotonic plastic zone ahead of the fatigue crack tip. The cubic strain is distributed along the grating, with monotonic plastic zone propagation at the initial and terminal part of the grating, at approximately a 30% ratio of the entire grating. However, the monotonic plastic zone is sensed by the FBG, at ±15% bias of the grating center, with the quadratic strain gradient pattern along the grating. In particular, when the initial and terminal parts of the grating experience highly inhomogeneous strain distribution, the spectrum distortion occurs.

Feasibility study of strain based stability evaluation of locally loaded slopes: Insights from physical and numerical modeling

Distributed strain monitoring of geotechnical structures has gained increasing attention in the past decade. The distributed fiber optic strain sensing (DFOSS) technology enables the measurement of strain distribution in soil slopes. This paper aims to investigate the feasibility of strain based slope stability evaluation for locally loaded slopes. The measurements of horizontal strains at different elevations in a two-dimensional (2D) model slope subjected to a vertical surcharge load were analyzed. Empirical relationships between different types of strain parameters and factors of safety calculated by the conventional method were established. To verify the above findings, a 2D finite element model of a homogeneous soil slope was built. By applying a gradually increasing local load on the numerical model, the strains in the soil mass and eventually slope failure were induced. The strain distributions of several virtual monitoring lines under different loading levels were captured and analyzed in detail. At the same time, the strength reduction method (SRM) was used to perform slope stability analysis. The results show that the strain distribution characteristics are closely related to the propagation of plastic zones and the formation of the critical slip surface. Taking into consideration the convenience of field instrumentation and monitoring sensitivity, the maximum strains at different elevations can be used as characteristic parameters for estimating the slope stability condition. Compared with conventional displacement based slope stability evaluation method, the proposed methodology is more efficient and sensitive, which makes full use of the benefits of the DFOSS technology.

Plasticity and strain localization around a horizontal wellbore drilled through a porous rock formation

Predicting plastic deformation and localization band around a wellbore drilled through a porous rock is a challenging task that could have important implications for the prediction of instability and sand production. This study focuses on analyzing stress field, plastic zones, and localized deformations around a horizontal borehole drilled at a great depth through a highly porous rock formation. Several laboratory studies demonstrate that, depending on the loading path, highly porous rocks are susceptible to different failure mechanisms, but most of these mechanisms are mainly due to shear-induced dilation and shear-enhanced compaction. Plasticity models, in conjunction with bifurcation analysis, represent an extremely useful framework for describing such detailed constitutive responses. This paper presents a new elastoplastic constitutive model characterized by two yield surfaces intersecting smoothly, that is able to capture the different failure modes. The model is validated against experimental data for several different porous rocks, and it is then used to determine the stress and strain distributions around a horizontal wellbore using nonlinear finite element analysis. Particular interest is devoted to predicting the condition for the formation of a localized band of intense deformation, elucidating the factors that either prevent or enhance the band initiation. Results of simulations show the key role played by the elastoplastic constitutive model and the effects of the mud pressure, the in-situ stress condition and geometric imperfections in the development and propagation of plastic zone, as well as in the initiation of localization zone.

Influence of periodic change in the yield strength in a plate on the development of plastic zones near a crack tip

An experimental study of plastic deformation zones in the vicinity of stress concentrators found a deviation of the shapes of plastic zones at the crack tip from the shapes obtained using conventional models. Moreover, the systems of bands of slip lines observed in the experiment and predicted by linear fracture mechanics are significantly different. Mathematical modeling of the propagation of plastic zones based on a finite element numerical solution of the equations of deformable solid mechanics was performed with the framework of the theory of large elastoplastic deformations. The basis of the mathematical model is the hypothesis that the structural heterogeneity of the material has a substantial effect throughout the volume of the test sample on the formation of plastic bands. Nonuniformity of the yield strength of the material is specified as a checkerboard distribution and as a system of horizontal and vertical bands. The results of the numerical calculations are compared with the experimental data, and it is shown that they are in qualitative agreement.

Experimental Analysis of Propagation of Plastic Zones in Two-Dimensional Problems

In the paper, the author presents experimental analysis of propagation of plastic zones in two-dimensional models with different stress concentrators. The experimental tests were carried out by photoelastic coating method on duralumin stripes loaded by tensile stresses. For various levels of loading, the photographs of isochromatic pattern were taken under loading and after removing loading. On the basis of isochromatic pattern recorded for loaded models, the boundaries of plastic zones were determined using the Treska-Coulomb yield condition. The isochromatic pattern taken for the unloaded, but previously partly plastified elements, show the picture of the residual strain remaining in the material. A discussion of the results is presented.

Elastic-Plastic Stress Analysis and Plastic Zone Propagation in Fibre-Reinforced Thermoplastic Laminates Using Finite Element Method

The results of a study of elastic-plastic stress analysis and plastic zone propagation in a low density polyethylene thermoplastic composite plate subjected to uniform transverse load are presented. Thermoplastic matrix, polyethylene (LDPE), and steel fibres were used to fabricate the composite plates. Yield points and elastic, elastic-plastic and residual stresses are calculated numerically for both symmetric and antisymmetric laminated plates with different ply orientations by using Finite Element Method. First order shear deformation theory was used as an equivalent single-layer laminate theory for small deformations. Contour plots corresponding to 100, 150 and 200 load steps display the expansion of the plastic zone. Mechanical properties of a layer were obtained experimentally.

Elastic–plastic behavior of woven-steel-fiber-reinforced thermoplastic laminated plates under in-plane loading

This paper deals with elastic–plastic stress analysis of thermoplastic laminated composite plates reinforced with woven steel fibers and containing circular holes under in-plane static loading. Residual stresses and propagation of plastic zones are obtained in simply supported symmetric and antisymmetric laminated composite plates by using a nine-node Lagrangian finite-element method. First-order shear deformation theory was used as an equivalent single-layer laminate theory for small deformations so that large plastic strains are not allowed to occur. A modified Newton–Raphson method is used as an iterative technique for determining the non-linear behavior of the material after yielding. After yielding the load is increased by means of many load steps, 0.01 MPa at each step, to follow the stress/strain curve more accurately. Mechanical properties of a layer were obtained experimentally.

Elastic–plastic beam-on-foundation under quasi-static loading

An elastic–plastic discrete spring model is developed to represent the mechanical behavior of an elastic–plastic beam-on-foundation (BoF) system, and an analytical procedure of analyzing the BoF under general quasi-static loading is formulated. The paper describes a detailed numerical simulation and analysis for the case of a BoF subjected to a concentrated force at the mid-span, and various plastic collapse mechanisms of BoFs are identified. Two peculiar phenomena, i.e. the migration of plastic hinge in the beam and the successive propagation of plastic zone in the foundation, are demonstrated. It is found that any elastic, perfectly plastic BoF system can be characterized merely by three non-dimensional parameters, but the limit state of a rigid, perfectly plastic BoF is determined by a single non-dimensional parameter only. The non-dimensional relative rigidity of BoF and the ratio of the maximum elastic deformation energies dissipated in the beam and foundation both play important roles in governing the deformation scenario of an elastic–plastic BoF system.

A dual homogenization and finite-element study on the in-plane local and global behavior of a nonlinear coated fiber composite

In order to uncover the influence of localized deformation in the soft interphase layer on the global elastoplastic behavior of a fiber-reinforced composite, a dual homogenization and finite-element study is presented in this paper. The homogenization approach makes use of the concept of secant moduli and an energy-based effective stress which is evaluated by a field fluctuation method. In the finite-element analysis NASTRAN was used, with the triangular elements carefully constructed in the unit cell to deliver the local stress and strain fields of the constituent phases. These dual investigations are complementary in that the homogenization theory could readily provide the overall stress–strain curves of the composite whereas the finite-element analysis could give the local stress fields which are important to the onset of damage or crack initiation. The results of finite-element method (FEM) could also be used to assess the accuracy of the homogenization theory. Detailed comparisons between the two have indicated that the homogenization theory is sufficiently accurate. The propagation of plastic zone in both ductile phases are also vividly demonstrated. In addition, the soft interphase is found to be a region of high strain concentration, rendering the entire system with a lower elastoplastic strength as compared to an ordinary 2-phase composite.

Rolling consolidation of metal matrix composites

Transverse rolling of metal matrix composite precursor wires is proposed as a consolidation technique for making sheets. Rolling in the transverse direction to the fiber orientation is analytically shown to be feasible, and longitudinal rolling results in fiber breakage. Plasticity analysis is conducted using Hill's general yield criterion for an isotropic materials and the associated Levy-Miles equations modified for plane strain conditions. The slab method is used to calculate the stresses in the material, and the effects of rolling parameters on the principal stress ratio are investigated. At the microscopic level, an elastic-plastic finite element formulation and a computation procedure are presented. Individual fibers are modeled to determine the stress state around each fiber. The principal stress ratio is suggested as a parameter that determines the tendency for void formation due to debonding and fiber breakage; finite element analysis is used to determine the effects of the principal stress ratio on the fiber-matrix interfacial stresses in the micromechanics model. The analysis determines the deformed mesh, plastic zone propagation and the stresses at the interface as a function of volume fraction and principal stress ratio. Interfacial stresses are assumed to be responsible for debonding during the deformation of metal matrix composites. This assumption and the results of the analysis provide guidelines for defining, the level of the biaxial stress field in the plane transverse to the fibers during rolling that will minimize interfacial fiber-matrix stresses.

Elasto-plastic analysis of an aluminum alloy matrix reinforced with silicon carbide fibers

The micromechanical analysis of a composite with a matrix of an aluminum alloy (Al 6061 T6) reinforced with fibers of silicon carbide was performed using the nonlinear finite element program CALSAP. The fibers were packed in a square array and the fiber volume fraction ( V/ O) was 36%. The matrix was assumed to have elasto-plastic material behavior, and the applied load was uniaxial tensile and transverse to the longitudinal axis of the fiber. The computation of the stress-strain distribution, the initiation of plastic yielding, the propagation of plastic zones and prediction of crack initiation within the matrix were performed. The results of the elasto-plastic analysis were compared with those obtained using the finite element program MARC and they are in close agreement.

Modélisation élastoplastique des coques minces par un élément courbe à variables nodales globales

ABSTRACT The purpose of this text is to present a geometrically (large displacements) and materially (plasticity) nonlinear analysis of thin plates and shells modelized by a curved triangular element based on Marguerre's theory. For computational efficiency an updated lagrangian description is used, referring the kinematical and mechanical field variables to local corotated coordinate frames. The curved shape of the shell is taken into account by Marguerre's theory. The constitutive equations are given in terms of increments of midsurface strains and curvatures and corresponding generalized stress (Eidsheim-Larsen approximation), thus avoiding the integration over the shell thickness and taking into account the propagation of plastic zones through the thickness. Integral approach is used to evaluate stress increments satisfying elastoplastic conditions, thus avoiding in some cases successive approximation process. The nonlinear response is determined using the Newton-Raphson method in conjunction with a s...

Borehole yield and hydraulic fracture initiation in poorly consolidated rock strata—part II. permeable media

The effective stress distributions around a plane strain circular opening in granular elastoplastic saturated permeable media are calculated. Plastic zone propagation is estimated by using a Mohr-Coulomb strain-weakening criterion, coupled with steady-state pore pressure distribution. Focusing on permeability changes around injection wells and considering factors such as cohesive strength, injection pressure and in situ stress, we investigate the effective stress evolution around the opening. The relation between the in situ stress and the hydraulic fracture initiation pressure is also investigated for a pre-yielded saturated media. It is concluded that the breakdown pressure cannot be used for estimating in situ stresses in such circumstances.

Elasto-plastic analysis of an aluminum alloy matrix reinforced with silicon carbide fibers packed in a hexagonal array

The micromechanical analysis of a composite with a matrix of an aluminum alloy (A1 6061 T6) reinforced with fibers of silicon carbide was performed using the nonlinear finite element program CALSAP. The fibers were packed in a hexagonal array and the fiber volume fraction (V/O) was 36%. The matrix was assumed to have elasto-plastic material behavior, and the applied load was uniaxial tensile and transverse to the longitudinal axis of the fiber. The computation of the stress-strain distribution, the initiation of plastic yielding the propagation of plastic zones, and prediction of crack initiation within the matrix were performed. The results of the elasto-plastic analysis were compared with those obtained using the finite element program MARC and they are in close agreement.