Plane Stress Case Research Articles

Elastic-Plastic Stress Analysis and Residual Stresses in a Woven Steel Fiber Reinforced Thermoplastic Composite Cantilever Beam Loaded Uniformly

In this study an elastic-plastic stress analysis is carried out in a woven steel fiber reinforced thermoplastic (LDFE, F.2.12) composite cantilever beam loaded uniformly at the upper surface. The intensity of the uniform force is chosen as a small value; therefore the normal stress component of ay is neglected during the elastic-plastic solution. An analytical solution is found, satisfying both equations of equilibrium and boundary conditions for a plane stress case. The orientation angle of the fibers is chosen as 00, 150, 300 and 45. The intensity of the residual stress component of ox is maximum at the upper or lower surfaces of the beam. Yielding begins earlier at the upper surface of the beam for 150 and 300 orientation angles. The intensity of the residual stress component of the shear stress is maximum on or around the longitudinal axis of the beam.

An elastic–plastic stress analysis of thermoplastic composite beams loaded by bending moments

In this study, an elastic–plastic stress analysis is carried out on a high-density thermoplastic-based composite cantilever beam loaded by a bending moment at the free end. The composite beam is reinforced unidirectionally by steel fibers, at 0°, 30°, 45°, 60° and 90° orientation angles. An analytical solution is performed for satisfying both the governing differential equation in the plane stress case and boundary conditions for small plastic deformations. The solution is carried out under the assumption of the Bernoulli–Navier hypotheses. It is found that the intensity of the residual stress component of σ x is maximum at the upper and lower surfaces or at the boundary of the elastic and plastic regions. The composite material is assumed to be strain-hardening. The Tsai–Hill theory is used as a yield criterion. The displacement components are found in the elastic region.

An implicit integration algorithm for plane stress damage coupled elastoplasticity

Boundary value problems based on Continuum Damage Mechanics(CDM) exhibiting damage with plasticity are highly nonlinear and usually, finite element method is employed for solution. The heart of a nonlinear finite element analysis is the integration of constitutive rate equations. An efficient and accurate integration scheme is required to integrate the rate equations. The basic aim of this paper is to present an efficient algorithm to integrate damage coupled elastoplasticity for plane stress case. It has been shown that the present integration algorithm is reduced to the linearLzation of three equations to solve them by Newton Raphson method unlike Doghri's algorithm[2] where fifteen equations are required to be linearized. Also, difficulties associated with extending the integration scheme of Simo and Taylor[9] to damage coupled elastoplasticity in plane stress case have been studied. Backward Euler scheme has been employed for the integration. An example has been presented to illustrate the efficiency of the algorithm.

Youngs modulus interpreted from plane compressions of geomaterials between rough end blocks

This note derives an approximate expression of the true Youngs modulus of a rectangular solid under plane compression between two rough end blocks, provided that the Poissons ratio ν of the solid is known. The friction between the loading platens and the ends of the specimen is assumed to be large enough to restrain slippage at the contact. By using the function space concept of Prager and Synge (1947) , a correction factor λ with calculable error is obtained which can be multiplied to the apparent Youngs modulus (i.e., the one obtained by assuming uniform stress field) to yield the true Youngs modulus; it is evaluated numerically for 0 ⩽ ν ⩽ 0.49 and 0 ⩽ η ⩽ 3 (where η = b⧸h with b and h being the half width and half length of the specimen) . In general, λ increases with ν and η for both plane strain and plane stress compressions. Within this range of ν and η, λ may vary from 0.37–1.0 for the plane strain case and from 0.84–1.0 for the plane stress case. Thus, the assumption of uniform stress field may lead to erroneous interpretation of the Youngs modulus. When the special case of ν = 1⧸3 and η = 1 is considered, we obtain λ = 0.9356, which compares well with 0.9359 obtained by Greenberg and Truell, 1948 .

Mechanical behavior of a μm-sized single crystal silicon structure with sharp notches

The mechanical behavior of μm-sized single crystal silicon structures is investigated in this paper. A new apparatus for microsample tensile testing is used to determine the mechanical properties of single crystal silicon microbars. Force and elongation are measured independently with high accuracy. Elastic constants as well as critical loads are determined. Failure of the sample occurs at the extremities of the microbar at a sharp notch. The stress field in the notch tip vicinity is therefore analyzed to characterize the load capacity of the structure. For this purpose, finite element computations are combined with approximate analytic solutions based on the Stroh formalism. Both the case of plane stress and two-dimensional displacements are considered. Comparison with the FEM solution shows that the plane stress assumption is better, leading to the determination of the critical stress intensity factor F cr for this case. Because F cr is dependent on the geometry, a new scalar failure criterion—i.e. a critical radius R cr—is introduced and derived from the calculated stress field at the notch. It is based on a comparison of stored elastic energy and surface energy at the critical location. Because it is a scalar criterion, it can be applied to single crystal microstructures with notches of arbitrary notch angle. For the case considered, experiments yield R cr = 0.8 nm. The proposed criterion can be considered as a step towards the definition of a design rule for sharp notches in single crystal structures, which is required for the optimized design of micromechanical devices.

Visualization of the plasticity-induced crack closure under plane strain conditions

A simplified dislocation model is used as an aid to visualize the developed plastic wedge under plane strain conditions. In contradiction to the plane stress case under plane strain conditions the plastic wedge is limited to the immediate vicinity of the crack tip. It is shown that the extra volume in the vicinity of the crack tip is caused by a plastic shearing of the material in the wake of the crack.

An Analytical Description of Slow Hysteresis in Polarized Ferroelectric Ceramic Actuators

Recently, an existing system of electroelastic equations was extended to account for slow hysteretic effects in polarized ferroelectric ceramic plates by employing an internal variable in a thermodynamic state function. All material irreversibility was taken to be a consequence of the ferroelectric polarization-electric field irreversibility. Since only the slowest possible hysteresis is of concern, the evolution equation is ignored and the known irreversible thickness polarization-electric field saturation curve is taken as the starting curve. Since all experimental data available to us are from strain measurements on a thin wide beam forced by voltages applied to very thin polarized ferroelectric ceramic actuators on the surfaces, the linear strain, plane-stress version of the equations is employed in the description of the actuator. The new irreversible material coefficients for the case of plane-stress are obtained from data on one major hysteresis loop along with the coefficients in the saturation polarization-electric field relations. These now known coefficients are then used to calculate other hysteresis loops, which are shown to agree quite well with measurement.

Ductility and toughenability study of epoxy resins under multiaxial stress states

The local strains in unmodified and rubber-modified epoxies under multiaxial stress states were examined. Matrix ductility was varied by using epoxide resins of different epoxide monomer molecular weights. The stress state was altered from a plane strain case to a plane stress case by varying the thickness of the test specimens. It was confirmed that, in the case of unmodified resins, the thinner specimens which experienced nearly uniaxial tensile stress exhibited much higher local strains at failure than the thicker counterparts which experienced highly triaxial tensile stress. Also, the cross-link density was reduced as monomer molecular weight increased, thus the increase in local plastic strain due to the stress state change also became greater. Furthermore, it was found that rubber modification markedly increased the plastic strain to failure, irrespective of the specimen dimensions, and that the extent of this plastic strain increased as cross-link density was lowered. These results are consistent with the concept that the cavitation of rubber particles relieves the initial multiaxial constraint in a thick specimen, induces a stress state closer to plane stress throughout the specimen, and consequently enables the matrix to deform to a larger extent. The results also show clearly that the toughenability of a matrix resin is not independent of the stress state and the matrix ductility.

FE calculation of elastodynamic energy release rate for rubbery materials

The elastodynamic energy release rate for rubbery material problems with a stationary crack under dynamic loads in the context of large deformation is calculated by the modified domain integral method with finite element solutions. General 2-D problems, including plane stress, plane strain, and axisymmetric cases, are considered in the formulation. The modified DIM is verified to be patch-independent. Although this property does not eliminate the need to include the singular near-tip area in. the calculation, it is observed in the numerical results that acceptable solutions of ERR are calculated even though an accurate near-tip model is not ensured.

Numerical evaluation of J 2 -integral for rubbery materials under dynamic loads with finite element method

The J2-integral for rubbery material problems with a stationary crack under dynamic large deformation is calculated with finite element solutions. General 2-D problems, including plane stress, plane strain, and axisymmetric cases, are considered in the formulation. The numerical procedure is verified to be path-independent in a generalized sense, and the near-tip region is hence always included in the calculation. A numerical treatment for simulation of the integration around the near-tip area is therefore proposed. It is demonstrated numerically that, with this treatment, the calculation is very insensitive to the crack tip finite element grid. In fact, no detailed information on the complicated asymptotic dynamic singular behavior for rubbery materials is required in the calculation.

Dynamic fracture of a kane-mindlin plate

We study dynamic crack problems for an elastic plate by using Kane-Mindlin's kinematic assumptions. The general solutions of the Laplace transformed displacements and stresses are first derived. Path independent integrals for stationary cracks subjected to transient loads and steadily growing cracks are deduced. For a stationary crack in a very thin plate subjected to impact loads, the crack tip dynamic stress intensity factor (DSIF), K 1( t), is related to the far field plane stress one, K 1 0( t), by K 1 ( t ) = K 1 0 ( t ) / 1 − v 2 where ν is Poisson's ratio. For a crack steadily growing with speed V, the crack tip DSIF, K 1( V), is given by K 1 ( V ) = B ( V ) / ( A ( V ) ( 1 − v 2 ) ) K 1 0 ( V ) where K 1 0( V) is the plane stress DSIF and A( V) and B( V) are known functions of V. These results are applied to compute the DSIF for a semi-infinite stationary crack in an unbounded plate subjected to impact pressure on the crack faces. The results of DSIF for a finite crack in an infinite plate under uniform impact pressure on the crack surfaces show that for each plate thickness, the maximum DSIF is higher than that for the plane stress case.

The Finite Analysis of Deformation and Stress Triaxiality of a Mixed I + II Mode Elastic-Plastic Crack Tip

The deformation and distribution of stress triaxiality σ_m/σ of mixed I + II mode elastic-plastic cracks under plane strain and plane stress conditions were analysed by the large strain finite element method and were compared with the HRR field. The results show that, (1) the model of crack-tip deformation under mixed I + II mode loading is sharpening-blunting, the sharpening and blunting deformation is acted on by compressive stress (σ_m/σ > 0) and tensile stress (σ_m/σ > 0), respectively; (2) in the plane strain case, the (σ_m/σ)_max of the real crack tip decreases with an increase of mode II component, but increases with an increasing strain hardening n for every mixed ratio. For the loading of K_I/K_II > 0.5, the (σ_m/σ)_max value of the real crack tip is less than that of the (σ_m/σ)_max given by the HRR field, but the opposite holds true cases where K_I/K_II > 0.5; (3) in the plane stress case, the (σ_m/σ)_max decreases a few values with an increasing mode II component, the values of (σ_m/σ)_max of every mixed ratio are far less than these of the corresponding loading condition in the plane strain case. In the plane stress case, the (σ_m/σ)_max values of every mixed ratio for any strain hardening exponent n are almost constant.

Higher order crack tip fields and its implication for fracture of solids under mode II conditions

Detailed results from an asymptotic mathematical analysis for the stress and strain fields at a crack tip embedded in a power-law elastic-plastic material under mode II loading conditions are presented. The asymptotic results include the higher order terms as well as the singular term, the HRR solution. Excellent agreement with the finite element numerical results confirms the analytical findings. It is further demonstrated that the HRR solution alone is able to characterize the stress and strain fields in a large area surrounding the crack tip which encompasses the critical distance controlling the fracture event. Therefore, it is anticipated that a single parameter such as J is sufficient to characterize the fracture event under pure mode II conditions for either plane stress or plane strain case. Mixed mode fracture is also discussed.

Application of the boundary element method to rubber-like elasticity

A boundary element method solution for the problem of finite deformation of rubber-like materials is presented. The solution is obtained by solving the nonlinear boundary element equations using an incremental-iterative procedure. The procedure is then implemented in a computer code that is tested through three numerical examples. In the first example, the plane strain problem of a thick-walled cylindrical pressure vessel is considered. In the second and third examples, which represent the plane stress case, a square rubber sheet with and without a central hole is subjected to finite plane deformation.

A formulation for the 4‐node quadrilateral element

AbstractA formulation for the plane 4‐node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second‐order Taylor series in the physical co‐ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices.For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix.The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non‐linear analysis. There, an application is given for the calculation of inelastic problems in physically non‐linear elasticity.The element is efficient to implement and it is frame invariant. Locking effects and zero‐energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.

Plane stress near-tip field analysis of steady-state crack growth along a linear-hardening elastic-plastic interface

In this work the asymptotic near-tip stress and velocity fields of a crack propagating steadily and quasi-statically along a ductile interface are presented for plane stress cases. The elastic-plastic materials are characterized by the J2-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power singularity in the radial distance to the crack tip. It is found that two distinct solutions exist with slightly different singularity strengths and very different mixities on the interface ahead of the crack tip. One of the solutions corresponds to a tensile-like mode and the other corresponds to a shear-like mode. An interface will change the near-tip fild of the tensile solution obviously, whereas the shear-like solution maintains its original structure as in homogeneous materials. In cases the elastic bimaterial parameter differs from zero, the two solutions can coalesce at some high strain-hardening. An interface between two high strain-hardening materials only slightly affects the stress and velocity distribution around the tip, whereas the singularity strength deviates from the homogeneous solutions. The strength of the singularity is predominantly determined by the smaller strain-hardening material. Poisson's ratio affects variation of the singularity as a function of strain-hardening slightly if the coalescing point of the variable-separable solution is not approached. Only for the very distinct elastic moduli the near-tip field approaches the rigid interface solution.

Stress-free edge effects on the transverse response of a unidirectional metal matrix composite

In using a representative volume element of a unidirectional composite, two out-of-plane boundary conditions represent edge and internal regions of a transversely loaded SCS-6/Timetal® 21S composite. A plane stress condition is proposed to represent a stress-free edge region of the composite, while a generalized plane (uniform longitudinal) strain condition is considered for internal regions. Numerical simulations with elastic-plastic matrix behavior and several different fiber-matrix interface strengths reveal a complex interaction of residual stress, fiber-matrix separation and matrix inelastic behavior which are all dependent on the out-of-plane boundary condition. Several permutations of plane stress and generalized plane strain solutions with several fiber-matrix interface strengths fail to accurately capture the non-linear behavior measured from experiments. The bounding of the experimentally determined transverse response between the plane stress and generalized plane strain solutions suggests that the transverse response may be a combination of both solutions. Strain measurements from the transversely loaded composite support speculation that the plane stress solution better represents edge regions of the composite than a generalized plane strain solution. Likewise, the generalized plane strain solution represents the internal strain state of the composite better than the plane stress case at low loads. Photomicrographs before and after transverse loading show fibers which have protruded from the edge of the specimen; thus, the strain state within the composite is transitioning to a less constrained condition.

Stress Analysis of an Orthotropic Work-Hardening Cylinder with Body Force*

An elastoplastic analysis of an axisymmetric cylinder subjected to linear body forces is presented. The effect of reinforcement and anisotropy are also included. Classical plasticity and familiar assumptions of plane stress and strain are used to arrive at closed-form solutions for the case of linear body forces. The problem is solved for the general case in which orthotropy is considered in the elastic range. For the case of plasticity, first the isotropic yield functions (von Mises and Tresca) are used and then the problem is extended to the case of Hill's yield criterion. Closed-form solutions are found for both the von Mises (plane strain) and Tresca (plane stress and strain) cases.

General steady creep analysis for various constitutive laws. Part 2: application and numerical examples in two dimensions

In the present paper, the general finite element method with penalty multiplier independent of the specific expression of the constitutive laws for steady creep presented in an earlier paper is applied to two-dimensional problems such as plane strain, plane stress and axisymmetry cases. In plane stress problems, it is found out that no penalty multiplier is needed. Several numerical examples are discussed to demonstrate the validity of this method. The significance of the reference stress method or R-node method has been shown in the approximate solution of the limit load. The reference stress can be obtained in two ways through several stress analyses of steady creep. One way is to perform these analyses by different constitutive laws. Another way is always to use the Norton law but with the exponent index taken as different values. The results of these two ways are consistent.

Notch and crack analysis as a moving boundary problem

An analysis of displacements and stresses around an elliptical notch in an infinite plate is presented. The analysis is based on linear-elastic material, but the problem is solved by taking into account the changes of geometry due to the applied load. Such a formulation makes it possible to study the change of the notch-tip geometry and its effect on the local stress and displacement fields during the process of quasistatic loading. It is found that in the case of a sharp notch, a significant change in the notch-tip radius takes place after the load is applied. This change results in a nonlinear variation of the stress concentration factor with applied load and the initial notch geometry. The solution obtained for a sharp crack yields finite stresses and strains at the crack tip, which becomes blunt soon after application of the load. Several closed form expressions are derived enabling the crack-tip radius, deformed crack shape and stresses to be easily calculated. All results are obtained for the plane-stress case.