Plane Stress Case Research Articles

Quasi-Static and Dynamic Crack Propagation by Phase Field Modeling: Comparison with Previous Results and Experimental Validation

In this paper, experimental tensile tests for pre-cracked high Carbon steel ‘C90’ specimens were performed for quasi-static and dynamic loading. High loading velocity affects the crack patterns by preventing deflection. On the other hand, an efficient numerical tool based on the phase field model was developed and validated to predict brittle fracture trajectories. A staggered numerical scheme was adopted to solve the displacement and damage fields separately. Implementation efficiency in initiating and propagating cracks, even from an undamaged microstructure, was proved. The effect of the critical fracture energy density Gc on the crack path was tested; with smaller Gc, the crack patterns become more complex. In addition, the impact of loading velocities was examined, and earlier and faster crack formation and greater crack branching is observed with higher impact velocity. In this study, bidimensional plane stress cases were treated. The phase field model with hybrid formulation was able to predict crack pattern and especially crack arrest and branching found in the literature. The developed model accurately determined the transition zone of the crack path topology that has been observed experimentally.

A Three-Dimensional Finite Element Analysis of A Two-Axially Pre-Loaded Plate Exposed To A Dynamic Force

In this paper, the forced vibration analysis by a harmonically time-dependent force of an elastic plate covered rigidly by a rigid half-plane is given. The plate layer is subjected to bi-axial normal initial force, into lateral sides separately. Here, the preloading state is exactly static and homogeneous. To eliminate the disadvantage of such a nonlinear model, the problem formulation is modeled in terms of the fundamental consideration of the theory of linearized wave in elastic solids under a pre-loaded state (TLWESPS) in a plane-stress case. For this purpose, considering Hamilton’s principles, the system of the partial equations of motion and the boundary-contact conditions are found. Based on the virtual work and the fundamental theorem of calculus of variation, the three-dimensional finite element method (3D-FEM) is used to understand the dynamic behavior of the plate. A numerical validation process is established. Next, influences of certain problem parameters such as Young’s modulus, aspect ratio, thickness ratio, pre-loaded parameter, etc. on the frequency mode of the pre-stressed system are given.

A novel constitutive model for S30408 stainless steel considering strain memory effect: Formulation and implementation

To predict seismic behavior of steel structures and conduct performance-based seismic design, a precise constitutive model to accurately reflect the elastic-plastic response of structural steel under various loading protocols is always needed, so that the strength, deformation capacity, and energy dissipation capacity of a steel component or system in case of a severe earthquake can be reasonably evaluated. On the other hand, the constitutive model should be convenient for practical engineering implementation and application. Previous studies show that certain type steel like stainless steel S30408 shows complex features under various strain ranges such as pronounced strain memory effect, unsaturated cyclic softening/hardening under small/large strain amplitudes, which differ from mild steel and low yield point steel. Therefore, the stainless steel S30408 is taken as the research object in this study to develop a new constitutive model to describe structural steels with such complex mechanical properties. First, a new nonlinear isotropic and kinematic hardening rule is extended based on the classical Chaboche model and a functional relationship between the hardening rate (defined as the variation value of peak stress in every adjacent cycle during cyclic loading) and the applied strain amplitude is established through a logistics function to describe the coupling effects. Second, the numerical implementation technique for three-dimensional is elaborated in detail. Third, the numerical implementation technique is extended to plane stress cases by the improved P projection method. Using the numerical integration algorithm, the proposed constitutive model can be successfully incorporated into ABAQUS/Standard through the UMAT subroutine interface which is suitable for both solid and shell elements.

Effect of out-of-plane dimension on simplified geometric models in stress analysis

Without exception, every physical object is three-dimensional. However, in many stress analysis situations the analyst is justified in using simplified two-dimensional idealizations of plane stress and plane strain, reducing the complexity of the problem. By obviating the need to mesh in the third dimension, this advantage also extends to numerical studies, helping economize significantly on time and computational power requirements. In plane stress idealization the out-of-plane stresses are zero, whereas in plane strain the out-of-plane strains are zero. These idealizations have variously been linked with the out-of-plane dimension, as well as the end conditions of the object under consideration. However, the exact correlation of the out-of-plane dimension with these idealizations remains ambiguous. Unlike the case of plane stress, there is much disagreement found in literature regarding the necessary conditions for the realization of plane strain, which needs to be addressed. In this article, finite element analysis was employed to study the effect of various out-of-plane dimensions and end constraints. The results show that there is no correlation between the out-of-plane dimension and plane strain, which depends only on the end constraints. However, the out-of-plane dimension does result in the transition from plane stress to generalized plane strain.

Deformations of the Varga material II: Plane stress

The governing equations for plane stress deformations of isotropic incompressible hyperelastic materials are highly nonlinear and consequently very few exact solutions are known. The Varga material is the only material in which general results are known. A recent paper by the authors revealed a linearization for the governing equations for the plane strain case of the Varga material. In this paper we show similar results for the plane stress case where we show the governing equations are also linearizable.

Nonlocal thermomechanical coupled modeling method for two-dimensional rolling contact using a peridynamic approach

The development of thermomechanical coupled simulations for rolling contacts is challenging when the adjacent material is temperature-dependent or when there are cracks or other discontinuities on the contact surface. This paper proposes a novel thermomechanical coupled modeling method for rolling contacts using a nonlocal peridynamic (PD) approach. Fully coupled, ordinary state-based PD thermomechanical equations for the plane strain and plane stress cases are derived based on the strain-energy density, including thermal coupling terms, and detailed information on each parameter is provided. Subsequently, an algorithm for rolling thermal contacts in PD is proposed based on the penalty method. This study further provides a detailed modeling process and an algorithm for the proposed method based on a multi-rate time-integration scheme for numerical implementation. A typical two-dimensional (2D) wheel–rail rolling contact model is established to evaluate the performance of the proposed contact modeling method in wheel–rail thermal contact and to analyze the influence of temperature-dependent material and rail-surface cracks on the thermomechanical response. It is found that the contact stress and rail-surface temperature rise in the case of small creepages and temperature-independent materials are in good agreement with the analytical solution. In the case of large creepages, the mechanical and temperature results of using temperature-dependent materials and temperature-independent materials are quite different, and the frictional heat generated can cause martensitic transformation of adjacent materials. Rail-surface cracks cause a sharp increase in the contact stress and temperature rise near the cracks, and the high temperature also promotes crack growth.

Direct stress minimization in electro‐mechanical metamaterials

AbstractMetamaterials are artificially created multiscale materials with many different applications [1]. We assume periodic microstructure. No specific length scale is demanded for the differentiation between the scales. Instead, we consider the microscale as consisting of repeating unit cells, whereas the macroscale is the scale, where desired and sometimes unusual physical effects occur. The precise arrangement and geometry at the microscale level is responsible for the macroscopic material behavior, which can substantially differ from the original components it is made from. This way, unique material properties otherwise not found in nature (e.g. negative refractive materials [2]) are possible. Most previous research regarding metamaterials concentrated only on a single physical branch in each case, e.g. electromagnetic or acoustic metamaterials [3,4].In this contribution we present a class of theoretical electro‐mechanical metamaterials by combining insulating and conducting materials. Our aim is to directly control and reduce the resulting total stress of insulating materials by counteracting the mechanical stress through the application of an electric field, which is created by a conducting material. The solution of the resulting minimization problem is related to the eigenvalues of the mechanical stress tensor. Additionally, we discuss the constrained cases of tension and compression and cover the plane stress case. We show numerical results for all cases and discuss the limits of such a material.

Двухпараметрический критерий разрушения для трещины нормального отрыва

The study introduces a two-parametric criterion for maximum tangential stresses adapted to the Mode I crack, which includes T-stresses. The difference of this criterion is that the size of the fracture process zone is written with account for the T-stresses, while the tangential stresses in the fracture process zone are equated to the local strength of the material rather than to the ultimate strength, as it is usually done. The paper gives two-parametric fracture criteria for the case of plane stress and strain states, allowing the strain constraint along the crack front. Within the study, we obtained expressions for the effective stress intensity factor which include the T-stresses — ultimate strength ratio, in addition to the stress intensity factor. This expression allows us to determine the most dangerous point of the crack front and, provided we know the fracture toughness of the material, to estimate the crack resistance of the part with the crack. As an example of using the fracture criterion, we considered a semi-elliptical edge crack in a plate stretched in two directions.

A hyperelastic extended Kirchhoff–Love shell model with out-of-plane normal stress: I. Out-of-plane deformation

This is the first part of a two-part article on a hyperelastic extended Kirchhoff–Love shell model with out-of-plane normal stress. We present the derivation of the new model, with focus on the mechanics of the out-of-plane deformation. Accounting for the out-of-plane normal stress distribution in the out-of-plane direction affects the accuracy in calculating the deformed-configuration out-of-plane position, and consequently the nonlinear response of the shell. The improvement is beyond what we get from accounting for the out-of-plane deformation mapping. By accounting for the out-of-plane normal stress, the traction acting on the shell can be specified on the upper and lower surfaces separately. With that, the new model is free from the “midsurface” location in terms of specifying the traction. We also present derivations related to the variation of the kinetic energy and the form of specifying the traction and moment acting on the upper and lower surfaces and along the edges. We present test computations for unidirectional plate bending, plate saddle deformation, and pressurized cylindrical and spherical shells. We use the neo-Hookean and Fung’s material models, for the compressible- and incompressible-material cases, and with the out-of-plane normal stress and without, which is the plane-stress case.

Inverse identification of large strain plasticity using the hydraulic bulge-test and full-field measurements

The hydraulic bulge test (HBT) is an important experimental technique to identify the properties of materials and, in particular, metals. One of the main advantages of such technique, in testing metals, is the possibility of reaching high levels of deformation before fracture. Moreover it allows the investigation of a biaxial loading condition. A hemispherical test can be performed using mechanical punches or the hydraulic pressure of a fluid. When hydraulic pressure is used, the test is not affected by friction. The hydraulic bulge test (HBT) is therefore a very efficient method to evaluate the properties of metals at large strains. Usually, the outcome of the HBT is the hardening curve in the equi-biaxial stress state that occurs at the top of the dome, during the expansion. In this paper a different approach is proposed, where the full-field displacement field obtained with stereo-DIC is used in an inverse identification scheme. The finite strain theory is used to map the stress and strain in the reference configuration so that the problem can be reduced to a 2D plane stress case. The method was first validated using a numerical model of the HBT and then applied on a real test performed on a BH340 steel.

Evaluation of Hoffman and Xia plasticity models against bi-axial tension experiments of planar fiber network materials

The anisotropic properties and pressure sensitivity are intrinsic features of the constitutive response of fiber network materials. Although advanced models have been developed to simulate the complex response of fibrous materials, the lack of comparative studies may lead to a dubiety regarding the selection of a suitable method. In this study, the pressure-sensitive Hoffman yield criterion and the Xia model are implemented for the plane stress case to simulate the mechanical response under a bi-axial loading state. The performance of both models is experimentally assessed by comparison to bi-axial tests on cruciform-shaped specimens loaded in different directions with respect to the material principal directions. The comparison with the experimentally measured forces shows the ability of the Hoffman model as well as the Xia model with shape parameter k≤2 to adequately predict the material response. However, this study demonstrates that the Xia model consistently presents a stiffer bi-axial response when k≥3 compared to the Hoffman model. This result highlights the importance of calibrating the shape parameter k for the Xia model using a bi-axial test, which can be a cumbersome task. Also, for the same tension-compression response, the Hill criterion as a special case of the Hoffman model presents a good ability to simulate the mechanical response of the material for bi-axial conditions. Furthermore, in terms of stability criteria, the Xia model is unconditionally convex while the convexity of the Hoffman model is a function of the orthotropic plastic matrix. This study not only assesses the prediction capabilities of the two models, but also gives an insight into the selection of an appropriate constitutive model for material characterization and simulation of fibrous materials. The UMAT implementations of both models which are not available in commercial software and the calibration tool of the Xia model are shared with open-source along with this work.

On pure bending in non-linear elasticity: A circular closed-form 2D solution for semi-linear orthotropic material

Large-deformation pure bending of a straight beam of uniform rectangular cross-section in plane strain is re-visited and generalised to an orthotropic linear-elastic material and the case of plane stress. The beam is bent into a completely shearless state by means of normal surface traction of a specific profile. For a linear-elastic material expressed in terms of Biot strain and stress tensors (orthotropic generalisation of the so-called semi-linear material) the solution still exists in closed form and is expressed in terms of the applied cross-sectional traction. As such, it provides a convenient benchmark reference for numerical methods in non-linear elasticity.In agreement with the isotropic case, through-the-thickness stresses develop in the form a hyperbolic cosine, but now with the magnitude actually depending on a measure of orthotropy in the material, while the longitudinal stresses change over the thickness in the form of a hyperbolic sine. In general, the unstrained axis is displaced from the centroidal axis.For an ideal material with infinite yield strength, the theoretical maximum of the magnitude of the longitudinal traction is limited to the amount of the geometric mean of the effective Young's moduli in the longitudinal and the through-the-thickness directions while, for a brittle material, bending strength reduces by approximately 715 of the square of the ratio between the magnitude of the applied traction and this mean value.

Analysis of Plane-Stress Systems via Total Potential Optimization Method Considering Nonlinear Behavior

As an alternative structural analysis method, total potential optimization using metaheuristic algorithms (TPO/MAs) has been demonstrated to be very effective, especially in nonlinear cases, for trusses and trusslike systems such as tensegric structures and cable networks. A recent study showed that TPO/MAs also can be applied successfully to plane-stress problems with linear constitutive equations. The present study enlarged the application area of TPO/MAs to the analysis of plates for plane-stress cases with nonlinear stress–strain equations. A relevant formulation was developed and used to solve five problems. Because TPO/MAs can be applied by using different metaheuristic algorithms, a number of them, some hybrid, were used, and they were compared among themselves and with finite-element solutions. The final example was a tunnel problem with 150 nodes, i.e., with 265 unknowns. This problem also was solved via hybrid algorithms. Several proposals for future research are listed.

Asymptotic path-independent integrals for the evaluation of crack-tip parameters in a neo-Hookean material

In this paper, we develop new asymptotic path-independent integrals for the evaluation of the crack tip parameters in a 2D neo-Hookean material. The new integrals are of both J-integral and interaction energy integral type and rely on the separation of the asymptotic boundary value problem into independent problems for each of the deformed coordinates. Both the plane stress and plane strain cases are considered. The integrals developed are used to compute the amplitude parameters of the asymptotic crack tip fields, which allows for direct extraction of these parameters from numerical results. A long strip with an edge crack under mixed loading modes is considered for both homogeneous and biomaterial cases. It is found that the asymptotic J-integrals produce good results for the first-order parameters while the interactions integrals produce good results for both the first and second-order parameters.

Semi‐infinite moving crack in an orthotropic strip sandwiched between two identical half planes

AbstractThe problem of semi‐infinite moving crack situated in a strip of a linear elastic orthotropic material sandwiched between two identical half planes has been considered. The assumed boundary value problem has been solved to convert it into the standard Wiener‐Hopf equation by using Fourier transform technique. Further the Wiener‐Hopf equation has been solved to obtain analytical expression for stress intensity factor and crack opening displacement at the tip of the crack for plane stress and plane strain cases. The effects of elastic material constants and propagation of the crack tip on stress intensity factor and crack opening displacement at the tip of crack have been displayed graphically for various particular cases for a combination of orthotropic materials.

A dynamic constitutive model for fiber-reinforced composite under impact loading

A constitutive model was proposed to predict the deformation and characterize the failure features of fiber-reinforced composite structures under dynamic loading. Primarily, quasi-static and dynamic mechanical tests were carried out on composite laminates with designed plying sequences. The material modulus was found rising as the increase of strain rate, while the ultimate strain exhibits no strain rate dependency. Nonlinear behavior was observed under transverse compression and in-plane shear, and was attributed to the progressive damage of material. Based on the results of uniaxial tests, constitutive relations were established for plane stress case and three-dimensional case respectively. Two damage factors were introduced to describe the nonlinear behavior, and failure criteria based on transversely isotropic invariants were proposed. Additionally, the constitutive model was implemented in Abaqus through user-defined material subroutine and validated by the simulation of uniaxial tests and bird impact tests. Comparing to the traditional model, the constitutive relations presented in this paper was proved to have better accuracy and efficiency in predicting dynamic response and ultimate failure patterns of composite structures under dynamic loading.

Stress concentration issues in unidirectional laminates

This paper analytically studies how elliptical holes affect the resistance of unidirectional laminate plates. Although local stress and strain concentration effects induced by notches are well known in isotropic materials, the same cannot be said about anisotropic materials. Stroh formalism is used to describe the elastic stresses around the border of elliptical holes subjected to in-plane loads. Tsai–Wu, Puck, and LaRC05 failure criteria are applied to estimate initial damage, due to their good performance on the world-wide failure exercise. Plane strain conditions are also studied to evaluate 3D constraint effects induced by the plate thickness. The major conclusions are: (1) the stress concentration may be up to about twice higher than for isotropic materials; (2) matrix failure is observed for most cases, and the notched strength is smaller than 10% of the unnotched strength; (3) strength prediction differences assuming plane stress and plane strain limit cases may be higher than 50%; and (4) for compressive loads, the LaRC05 model predicts a considerable strength reduction when compared to the other failure criteria.

Effect on horizontal pressure in steel silos evoked by a sudden change in the ambient temperature

The Eurocode EN 1991-4 includes provisions which refer to the stress increase in structural members evoked by the temperature differential between the silo wall and the stored material. This Eurocode presents in the clause 5.6 the formula on increase of horizontal pressure as a result of the temperature differential between the silo wall and the stored material. The derivation of this formula by two different approaches was presented in this work. Theoretical considerations were exemplified by chosen examples showing the effort increase of the silo wall as a result of a sudden cooling of the silo wall. Silos of different slenderness and of different coefficients of wall friction were considered. The performed analysis has revealed that in every case the formula based on the plane strain assumption gave higher values of the horizontal pressure increase. Calculations have revealed that the sudden temperature drop of the silo wall by 20 °C can lead even to the 35 % increase of horizontal pressure. Results of analytical solutions were supplemented by numerical simulations which have confirmed correctness of the formulae derived in this work. The alternative formula based on the plane strain assumption should replace the recommended in Eurocode formula based on the plane stress case.

Anisotropic finite strain viscoelasticity: Constitutive modeling and finite element implementation

A new anisotropic finite strain viscoelastic model is presented, which is based on the Holzapfel type anisotropic hyperelastic strain-energy function. The anisotropic viscous part is set to be independent from the isotropic viscous part. A corresponding multiplicative decomposition of the deformation gradient is presented, and a specific definition of the anisotropic viscous fiber term. A new method to develop the evolution equations of the viscous internal variables is also provided. The time derivatives of the internal variables for the isotropic and anisotropic viscous parts are obtained from the evolution equation of the second Piola–Kirchhoff stress for the viscous part. The corresponding analytical validation of non-negative dissipation using the second law of thermodynamics is provided. The incompressible plane stress case is used to achieve an analytical solution for the proposed constitutive model. A good agreement between the finite element results and the analytical solution is obtained. Finally, some numerical simulations are presented, including the viscous hysteresis response, experimental data fitting and a relaxation test.

Inverse determination of elastic constants of a hyper-elastic member with inclusions using simple displacement/length measurements

There are many engineering parts and natural soft members, which consist of several phases with different mechanical properties. Identification of material parameters of multi-material members is an important issue that has attracted considerable attention. In this article, a finite-element-based inverse method for identification of material parameters of a body made of several hyper-elastic materials is presented. In the presented inverse method, the unknowns are computed using a few simple displacement/length measurements in a few elasto-static experiments. The inverse analysis can be carried out without any full-field measurements. Both plane strain and plane stress conditions are considered. In the plane strain case, displacements of several sampling points on the boundary of the problem domain are used as measured data. In the plane stress case, some characteristic lengths of the member in the deformed configuration are used as measured data. A cost function in terms of the differences between calculated and measured values is defined and the Gauss–Newton method is used to minimize it. The multi-material sensitivity analysis in the optimization process is made by analytical differentiation of the weak form of the problem. Both numerical and experimental studies are made to demonstrate the effectiveness and accuracy of the proposed methods.