Finite Deformation Theory Research Articles

Plastic theory for the multi-crystal metals —From infinitesimal deformation to finite deformation

Abstract Multi-crystal metals have the property of volume conservation in the plastic state. In the infinitesimal deformation plasticity the strain tensor can be split into a deviator part and a volumetric part. The vanishing of the first variant of the strain tensor is equivalent to the volume conservation. Furthermore, the split of the strain into an elastic part and a plastic part is also adopted widely. The flow rule is thus established. These two splits are not confirmed in the finite deformation plasticity. The plasticity criterion and the flow rule are thus facing great challenge. There are various definitions of strain measures in the finite deformation theory. Though the choosing of strain measure is arbitrary in the elastic problem, it is strongly restricted in the plastic problem. By theoretical and experimental studies, it is shown that the logarithmic strain is the only suitable strain measure in the metal forming problem.

Modeling mechanical response of intumescent mat material at room temperature

Intumescent mat material is widely used to support ceramic substrates in catalytic converters and behaves very much like hyper-foam material under compressive loading. Experiments show that compressive loading curves depend on the ram speed and the number of cycles. The unloading curves show different slopes and paths that depend less on the ram speed and number of cycles. The slopes of the unloading curves decrease as the plastic strain increases; this is referred to as “softening” in this study. The effects of rate, softening, and plastic deformation must be considered to model the mechanical response of intumescent mat material. Finite deformation theory is applied with a multiplicative decomposition of the deformation gradient tensor. The developed theory is implemented as an implicit finite element algorithm in ABAQUS TM/STANDARD. The necessary material parameters are extracted from experiments. Numerical simulations show good agreement with experiments.

Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field

A finite deformation theory of mechanism-based strain gradient (MSG) plasticity is developed in this paper based on the Taylor dislocation model. The theory ensures the proper decomposition of deformation in order to exclude the volumetric deformation from the strain gradient tensor since the latter represents the density of geometrically necessary dislocations. The solution for a thin cylinder under large torsion is obtained. The numerical method is used to investigate the finite deformation crack tip field in MSG plasticity. It is established that the stress level around a crack tip in MSG plasticity is significantly higher than its counterpart (i.e. HRR field) in classical plasticity.

Bifurcation of elastoplastic solids to shear band mode at finite strain

This paper formulates and implements a finite deformation theory of bifurcation of elastoplastic solids to planar bands within the framework of multiplicative plasticity. Conditions for the onset of strain localization are based on the requirement of continuity of the nominal traction vector, and are described both in the reference and deformed configurations by the vanishing of the determinant of either the Lagrangian or Eulerian acoustic tensor. The relevant acoustic tensors are derived in closed form to examine the localization properties of a class of elastoplastic constitutive models with smooth yield surfaces appropriate for pressure-sensitive dilatant/frictional materials. A link between the development of regularized strong discontinuity and its unregularized counterpart at the onset of localization is also discussed. The model is implemented numerically to study shear band mode bifurcation of dilatant frictional materials in plane strain compression. Results of the analysis show that finite deformation effects do enhance strain localization, and that with geometric nonlinearities bifurcation to shear band mode is possible even in the hardening regime of an associative elastoplastic constitutive model.

FE modeling of thermo‐elasto‐plastic finite deformation and its application in sheet warm forming

In this paper, a strategy for analyzing a problem of the transient thermal coupling with the elastoplastic finite deformation is presented. A general constitutive equation is deduced by assuming the material properties to be temperature‐dependent. The thermal and mechanical coupling problem is solved with a staggered algorithm, which partitions the coupled problem into an elasto‐plastic problem at the known temperature field and a pure heat transfer problem at the fixed configuration. In this procedure, the elasto‐plastic mechanical analysis is based on the static‐explicit solution algorithm, which applies the finite deformation theory to describe the nonlinear behavior of the deformation body and its contact interaction with the tools during the forming process induced by the ordinary external loading and the “thermal loading”. In addition, both the ordinary heat transfer boundary conditions and the mechanical terms are taken into account in the implicit finite element analysis of the heat transfer. A special method based on the R‐minimum strategy is presented to solve the interaction problem between the static‐explicit mechanical analysis and the implicit thermal analysis. Furthermore, as examples, the analyses of sheet warm forming processes are demonstrated.

Passive Biaxial Mechanical Properties of the Rat Bladder Wall After Spinal Cord Injury

Changes in the mechanical properties of the bladder wall after spinal cord injury can alter bladder compliance and wall tension, leading to changes in afferent nerve activity and to abnormal reflex mechanisms. To our knowledge we report the first application of biaxial mechanical testing to the normal bladder wall and demonstrate how these properties change after spinal cord injury. Whole bladders were harvested from mature female Sprague-Dawley rats weighing 250 to 300 gm. Test group animals underwent complete spinal cord transection at the T9 to T10 level and normal animals comprised the control group. The bladders were cut open longitudinally, the trigone and apex were removed and the remaining tissue was trimmed to a square of 9 to 13 mm. per side. Mechanical properties of the tissue sample were tested using planar biaxial testing, in which a stress was applied in the circumferential and longitudinal (base-apex) directions, and resulting axial strains were measured. In normal and spinal cord injured rats bladder wall tissue demonstrated isotropic mechanical behavior when equal stress levels were applied in anatomical directions. However, under nonequi-biaxial loading bladder specimens were not truly isotropic but displayed anisotropic-like behavior. Spinal cord injured tissues were consistently more compliant than normal controls. Biaxial mechanical testing can detect differences in normal control and hypertrophied rat bladders 10 and 14 days after spinal cord injury. These changes represent an important component of the bladder response to spinal cord injury.

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.

Wrinkling of tubes in bending from finite strain three-dimensional continuum theory

Abstract The problem of a tube under pure bending is first solved as a generalised plane strain problem. This then provides the prebifurcation solution, which is uniform along the length of the tube. The onset of wrinkling is then predicted by introducing buckling modes involving a sinusoidal variation of the displacements along the length of the tube. Both the prebuckling analysis and the bifurcation check require only a two-dimensional finite element discretisation of the cross-section with special elements. The formulation does not rely on any of the approximations of a shell theory, or small strains. The same elements can be used for pure bending and local buckling a prismatic beam of arbitrary cross-section. Here the flow theory of plasticity with isotropic hardening is used for the prebuckling solution, but the bifurcation check is based on the incremental moduli of a finite strain deformation theory of plasticity. For tubes under pure bending, the results for limit point collapse (due to ovalisation) and bifurcation buckling (wrinkling) are compared to existing analysis and test results, to see whether removing the approximations of a shell theory and small strains (used in the existing analyses) leads to a better prediction of the experimental results. The small strain analysis results depend on whether the true or nominal stress–strain curve is used. By comparing small and finite strain analysis results it is found that the small strain approximation is good if one uses (a) the nominal stress–strain curve in compression to predict bifurcation buckling (wrinkling), and (b) the true stress–strain curve to calculate the limit point collapse curvature. In regard to the shell theory approximations, it is found that the three-dimensional continuum theory predicts slightly shorter critical wrinkling wavelengths, especially for lower diameter-to-wall-thickness ( D / t ) ratios. However this difference is not sufficient to account for the significantly lower wavelengths observed in the tests.

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.

Multiple decomposition in finite deformation theory

A strain decomposition method is proposed in finite strain deformation theory. The method is based on the multiplicative decomposition of the deformation gradient with the assumption of intermediate configurations. Kinematically correct additive decomposition of the strain is developed. The strain and stress measures are calculated by way of the dual variables. Geometric linear consitutive models are generalized to finite strain theory. An application and an example are also included for thermoelastoplastic analysis.

A study of oblique cutting for different low cutting speeds

In this paper, the finite deformation theory and an updated Lagrangian formulation (ULF) were used to describe the oblique cutting process. Either the tool geometrical location condition or the strain energy density constant was combined with the twin node processing method to be adopted as the chip separation criterion. An equation for 3D tool face geometrical limitation was established to inspect and correct the relation between the chip node and tool face. In addition, a 3D finite difference equation for heat transfer was derived. Based on this approach, a coupled thermo-elastic–plastic large deformation finite element model for oblique cutting was established, for which mild steel was used as the workpiece material and P20 as the tool. Under the different cutting speed conditions, the chip deformation process and the effect of different cutting speeds on the chip flow angle, cutting force and specific cutting energy were first explored. Then, the effect of different cutting speeds on the separation location of the chip node and the geometrical phenomenon at the instant of chip separation from the tool face, and on both stress and temperature distributions on the chip surface, were analyzed. Finally, the effect of different cutting speeds on the residual stress, displacement and temperature distributions on the machined surface after cutting were investigated to understand the relation between the cutting speeds and the integrity of the machined surface. During the chip deformation process, the simulated chip flow angles under the different low cutting speed conditions approximately matched with the designated tool inclination angle, which complied with the geometrical requirements of Stabler’s criterion. Further, the simulated specific cutting energy under a given low cutting speed condition was compared with the experimental data, the result of which was within an acceptable range, and the trend of specific cutting energies under the different low cutting speed conditions were the same as the experimental trends. It is obvious from the above findings that the model presented in this paper is consistent with the geometrical and mechanics requirements, which verifies that the proposed model is acceptable.

Inverse Design Methodology of a Tire

Abstract Most rubber components usually suffer large deformation, thus one should consider finite deformation theories when designing rubber products such as tires. Although finite element analysis (FEA) is useful for computing nonlinear structural response in standard problems, it is more difficult to predict the undeformed original to-be-manufactured shape of a part corresponding to a design constraint involving a prescribed deformed shape under a given load. For example, one can use an optimization technique to predict the original shape in the sense of an inverse problem via successive iteration. As another procedure to solve such inverse problems, Govindjee and Mihalic [4,5] have proposed a new computational procedure to predict undeformed (to-be-manufactured) shapes for prescribed exterior deformed configurations, Cauchy tractions and displacement boundary conditions. In this paper, we demonstrate applications of the inverse shape determination problem for a tire design. The proposed methodology enables us to predict an undeformed (to-be-manufactured) tire shape corresponding to the design constraint of a prescribed exterior deformed configuration. After reviewing the formulation of the inverse shape determination, some numerical examples illustrating the methodology in a tire design are presented.

Equation of state of rare-gas crystals near their metallization

A nonempirical equation of state of a crystal is found in the framework of the finite-deformation theory. The adiabatic potential is calculated from first principles by using a set of localized functions which are exactly orthogonalized to one another, with the orthogonalizing matrix being calculated by the cluster expansion method. The most essential part of the equation of state which corresponds to short-range repulsion involves no experimentally determined parameters. Comparison of the theory and experimental data in the range of large compressive deformations shows that the terms of higher orders in the overlap integral are of importance for neon, whereas it suffices to use the quadratic approximation for xenon. The reason for this is discussed.

410 共回転定式化による圧延加工の有限変形3次元弾塑性FEM解析(OS 塑性加工)

Three-dimensional FEM analysis for rolling processes is developed. It is based on finite deformation theory using co-rotational formulation. Three-dimensional FEM analyses of cold rolling processes are requested to predict 'shape' or flatness of rolled sheet, then the precise three-dimensional analyses of residual stresses are needed. Present report describes the method we used for developing an elastic-plastic code for analyzing such rolling processes, and several results of simulations are shown and discussed.

Microscopic Symmetric Bifurcation Analysis for Cellular Solids Based on a Homogenization Theory of Finite Deformation. 2nd Report. Application to In-Plane Buckling of Hexagonal Honeycombs.

This report deals with numerical verification of the theory, developed in the first report, on microscopic symmetric bifurcation of cellular solids subjected to macroscopically uniform compression. The theory is applied to analyzing the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is shown that microscopic symmetric bifurcation takes place in the heneycomb in spite of the complex buckling behavior under in-plane biaxial compression. It is also shown that the complex buckling behavior is a consequence of the multiplicity of buckling modes ; for example, a triplet of buckling modes under equi-biaxial compression, each of which is relatively simple, are linearly combined to generate a flower-like complex mode, which was observed recently to occur in a hexagonal honeycomb with circular cells.

Microscopic Symmetric Bifurcation Analysis for Cellular Solids Based on a Homogenization Theory of Finite Deformation. 1st Report. Theory.

In this paper, we analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, showing the principle of virtual work for periodic solids in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state the following postulate : At a microscopic symmetric bifurcation point, microscopic displacement rate gets spontaneous, but changing the sign of the spontaneous displacement rate field has no influence on the variation of macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the bifurcation point. The resulting conditions are discretized using a finite element method in order to employ the present theory in computational analysis. The finite element discretization also deals with a general case, which includes microscopic non-symmetric bifurcation as well as symmetric one.

An analysis of chip breaker tool effect on an oblique cutting process using a coupled thermo-elasticplastic FEM

The finite deformation theory, an updated Lagrangian formulation (ULF), a 3D finite difference equation for heat transfer, and a two-stage geometrical equation for an obstruction-type chip breaker tool face were used to describe the oblique cutting process. Thus, a coupled thermo-elastic-plastic large deformation finite element model for oblique cutting with an obstruction-type chip breaker tool was established. Mild steel was used for the workpiece, tool as P20 and the cutting speed was 274.8 mm/s. The chip deformation process, the variations of the cutting force, specific cutting energy, both stress and temperature distributions on the chip surface and residual stresses on the machined workpiece surface were analysed for different chip breaker lengths. The findings indicate that the shorter the chip breaker length, the lower the values of the aforementioned physical properties and the better the effect of chip breaking.

The numerical approach for determining the crack-tip strain rate in the estimation of crack growth rate

The finite deformation theory, an updated Lagrangian formulation (ULF), a 3D finite difference equation for heat transfer, and a two-stage geometrical equation for an obstruction-type chip breaker tool face were used to describe the oblique cutting process. Thus, a coupled thermo-elastic-plastic large deformation finite element model for oblique cutting with an obstruction-type chip breaker tool was established. Mild steel was used for the workpiece, tool as P20 and the cutting speed was 274.8 mm/s. The chip deformation process, the variations of the cutting force, specific cutting energy, both stress and temperature distributions on the chip surface and residual stresses on the machined workpiece surface were analysed for different chip breaker lengths. The findings indicate that the shorter the chip breaker length, the lower the values of the aforementioned physical properties and the better the effect of chip breaking.

Constructing the Nonlinear Small-Deformation Theory in the Mechanics of Deformable Bodies

The simplifications introduced into finite-deformation theory to reduce it to small-deformation theory are analyzed as applied to the nonlinear mechanics of deformable bodies. A version of the generalized small-deformation theory is considered

The finite deformation theory for beam, plate and shell. Part V. The shell element with drilling degree of freedom based on Biot strain

Due to the facts that the spatial moment force is non-conservative and the three components of the finite rotation pseudo vector are not independent, the ordinary variational principle, based on a functional achieving its stationary value, is not suitable for the problem subject to non-conservative loading. The finite element method proposed in this paper is based on the virtual work principle, rigid normal assumption, polar decomposition and total Lagrangian formulation.