Plastic Work Rate Research Articles

An energy-based isotropic compression relation for cement-admixed soft clay

This note presents an energy-based isotropic compression relationship for cement-admixed soil. By assuming that the rate of destructuration is proportional to the rate of plastic work done during loading for a given soil, a relationship between the structure and the change in specific volume is derived. An approximate relationship is also derived for the case when the swelling index of the structured soil is small. By using experimental data, the main assumption introduced for deducing the relationship is validated.

Effect of the Flow Rule on the Bearing Capacity of Strip Foundations on Sand by the Upper-Bound Limit Analysis and Slip Lines

The influence of the flow rule on the bearing capacity of strip foundations placed on sand was investigated using a new kinematic approach of upper-bound limit analysis. The method of stress characteristics was first used to find the mechanism of the failure and to compute the stress field by using the Mohr-Coulomb yield criterion. Once the failure mechanism had been established, the kinematics of the plastic deformation was established, based on the requirements of the upper-bound limit theorem. Both associated and nonassociated plastic flows were considered, and the bearing capacity was obtained by equating the rate of external plastic work to the rate of the internal energy dissipation for both smooth and rough base foundations. The results obtained from the analysis were compared with those available from the literature.

New method of determination of limit plastic load in intersecting cylindrical shells

In this paper, the method of determination of limit plastic load in intersecting cylindrical shells using maximum relative plastic work rate criterion is discussed. Numerical elastic-plastic analysis was performed using finite-element method, shell theory, and plastic theory. The calculated limit plastic moment values are compared with the known experimental and calculated data for six prototype vessel models having a radial branch pipe loaded by a moment in the longitudinal and cross-sectional plane of the joint.

J031011 応力増分方向依存性構成式の塑性異方性への展開

It is proposed the expression of a newly extended plastic constitutive model to initial anisotropy materials. Proposed flow rule permits a plastic strain rate tensor to follow up a certain extent a stress rate tensor which rotates instantly from the direction of current stress tensor as is the case of plastic instability phenomenon. It has quite same structure of the non-associated normality model in which the plastic potential function is defined independently from yield function. It is also suggested the expression of the equivalent plastic strain iate which is conjugated with the equivalent stress corresponding to the proposed yield function in plastic work rate. This is important to express generalized work-hardening rule for materials which have plastic flow stress anisotropy. The proposed theory may overcome the serious problems in the associated plastic flow theory.

Slip Deformation Analysis Based on Full Constraints Model for α-Titanium Alloy at Low Temperature

The effects of restricted slip conditions on both the Taylor factor and plastic work rate under the condition of tensile yielding have been analyzed in � -titanium alloys at low temperatures, using the full constraints model. The role of secondary slip systems, i.e., the hai basal slip and hc þ ai pyramidal slip, was clarified, when the hai prismatic slip was dominant. Although no influence of secondary slip conditions on the Taylor factor was detected, the plastic work rate was sensitive to the operating secondary slip systems. When the basal system was chosen as the secondary slip system, the plastic work rate increased in all tensile axes, especially around h0001i. In addition, no basal slip operation decreased the plastic strain energy. The plastic work rate was the highest along the h0001i tensile axis, and the operation of the hc þ ai pyramidal slip was necessary to achieve plastic deformation along c axis. High elastic strain energy, therefore, must accumulate to a high level around h0001i, because the pyramidal slip is hardly active owing to its very high critical resolved shear stress. [doi:10.2320/matertrans.L-M2011815]

Derivation of Plastic Specific Work Rate for Equal Area Yield Criterion and Its Application to Rolling

On the basis of the analysis of the hexagons inscribed and circumscribed of the Mises yield circle on the π-plane in Haigh Westergaard stress space, a new equation called equal area linear yield criterion or EA criterion was proposed, and its plastic specific work rate was derived. Then, a generalized worked example of plate rolling was analyzed by the criterion and its corresponding plastic specific work rate. Finally, the precision of the solution was compared with measured roll separation force in plate finish rolling process. It indicates that the calculated results by EA criterion are slightly higher than the measured ones.

Large Plastic Deformations Accompanying the Growth of an Elliptical Hole in a Thin Plate

Within the context of plane stress assumptions and approximations, an analytical solution is derived for the finite deformation of a traction-free elliptical hole in an infinite plate with tensile tractions at infinity. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under a deformation theory of plasticity. The governing partial differential equations are parabolic in nature and consequently have a single family of mathematical characteristics or slip lines associated with them. Each particle of mass follows a rectilinear path in the plane defined by its slip line which emanates orthogonally from the elliptical hole. By assuming a constant speed for each particle in the plane, a state of plane equilibrium is realized. The originally elliptical hole expands in the shape of an oval which is a parallel curve to the original ellipse. The slip lines remain orthogonal to the evolving oval hole as a necessary condition for a traction-free interior boundary. This solution also satisfies the material stability criterion that the rate of plastic work be positive throughout the entire body for all time. As this solution has some features associated with large deformation crack problems at elevated temperatures, possible applications include secondary or steady-state creep.

On scale invariance in anisotropic plasticity, gradient plasticity and gradient elasticity

A simple but robust scale invariance argument earlier used to derive flow rules for macroscopic kinematic hardening plasticity from the configuration of single slip is now applied to consider yielding of fiber reinforced composites as first pioneered, on a purely phenomenological basis, by Tony Spencer and his co-workers. It is shown that the approach results to simpler constitutive equations than those obtained by the phenomenological methodology of the standard representation theorems for isotropic functions. The argument is then extended to obtain simplified but physically-based constitutive equations for gradient plasticity and gradient elasticity as such type of approaches have been extensively adopted in the recent literature to interpret interesting phenomena at the micron scale, not captured by classical theories. For gradient plasticity, the scale invariance argument involves two steps: first the algebraic elimination of the orientation tensors defining crystal slip between the microscopic expressions for the back (dislocation) stress and plastic strain rate; and, second, a maximization procedure for the rate of plastic work (maximum dissipation or entropy production). For gradient elasticity the algebraic elimination of the lattice orientation vectors between the microscopic expressions for the lattice stress and lattice strain suffices, as no dissipation occurs in this case. As an application of gradient theory, we consider, in particular, the size effect in strain gradient-dependent plastic layers and the removal of elastic singularities at screw dislocation lines and crack tips under a shear mode loading are considered. It is shown that a number of insightful results are obtained which are in accord with experimental observations and/or elaborate numerical simulations. This is largely due to the fact that the method used provides simple expressions for the strain/stress field which turns out to be governed by a non-homogeneous Helmholtz equation for which analytical solutions are available.

Deduction of plastic work rate per unit volume for unified yield criterion and its application

A unified linear expression of plastic work rate per unit volume is deduced from the unified linear yield criterion and the associated flow rule. The expression is suitable for various linear yield loci in the error triangle between Tresca's and twin shear stress yield loci on the π-plane. It exhibits generalization in which the different value of criterion parameter b corresponds to a specific linear formula of plastic work rate per unit volume. Finally, with the unified linear expression of plastic work rate and upper-bound parallel velocity field the strip forging without bulge is successfully analyzed and an analytical result is also obtained. The comparison with traditional solutions shows that when b=1/(1+√3) the result is the same as the upper bound result by Mises' yield criterion, and it also is identical to that by slab method with m=1, σ 0=0.

Research on the Plastic Work Rate Consumed in the Process Zone for the Growth of Triaxial-Stress-State Crack

When studying the growth of triaxial-stress-state crack, the plastic work rate consumed in the process zone at crack tip should be described clearly. Fracture for a cracked specimen is related to rupture of a simple uniaxial tensile specimen by nature. By equaling the crack tip stress-strain field to uniaxial stress-strain field, it is obtained that the plastic work rate consumed in the process zone at crack tip can be expressed by the plastic energy density WF multiplied by the equivalent size of the process zone hF. The plastic energy density WF can be expressed by virtue of uniaxial true stress-strain curve. And analytical form of hF for triaxial stress state is presented by recourse to three-dimensional analyses on material behaviors. The expression of plastic work rate consumed in the process zone is verified by the predicting fracture toughness.

Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials

abstract In the present paper we consider for a < x < b, 0 < t < T, the system of partial differential equations $$ \begin{array}{l} \displaystyle{\rho(x) {\partial v \over \partial t} - {\partial \over \partial x} \left(\mu(x,\theta) {\partial v \over \partial x}\right) = f,} \displaystyle{c(x,\theta) {\partial\theta \over \partial t} = \mu(x,\theta) \left({\partial v \over \partial x}\right)^2}, \end{array} $$ completed by boundary conditions on v and by initial conditions on v and θ. The unknowns are the velocity v and the temperature θ, while the coefficients ρ, μ and c are Caratheodory functions which satisfy $$ 0 < c_1 \leq \mu(x,s) \leq c_2, \quad {\partial\mu \over \partial s}(x,s) \leq 0, $$ $$ 0 < c_3 \leq c(x,s) \leq c_4,\quad 0 < c_5 \leq \rho(x) \leq c_6. $$ This one dimensional system is a model for the behaviour of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and temperature dependent rate of plastic work converted into heat. Under the above hypotheses we prove the existence of a solution by proving the convergence of a finite element approximation. Assuming further that μ is Lipschitz continuous in s, we prove the uniqueness of the solution, as well as its continuous dependence with respect to the data. We also prove its regularity when suitable hypotheses are made on the data. These results ensure the existence and uniqueness of one solution of the system in a class where the velocity v, the temperature θ and the stress $\sigma = \mu(x,\theta) \displaystyle{\partial v \over \partial x}$ belong to L∞((0,T) × (a,b)). Keywords: Thermoviscoplastic materials, nonhomogeneous materials, thermal softening, existence, uniqueness, Galerkin’s method Mathematics Subject Classification (2000): 74H20, 74H25, 65M60, 35D05, 35D10, 35R05, 74C10, 74F05, 35Q72, 35M20 This is a “Springer Open Choice” article. Unrestricted non-commercial use, distribution, and reproduction in any medium is permitted, provided the original author and source are credited.

A Complete Perfectly Plastic Solution for the Mode I Crack Problem Under Plane Stress Loading Conditions

A continuous stress field for the mode I crack problem for a perfectly plastic material under plane stress loading conditions has been obtained recently. Here, a kinematically admissible velocity field is introduced, which is compatible with the continuous stress field obtained earlier. By associating these two fields together, it is shown that they constitute a complete solution for the uncontained plastic flow problem around a finite length internal crack, having a positive rate of plastic work. The yield condition employed is an alternative criterion first proposed by Richard von Mises in order to approximate the plane stress Huber-Mises yield condition, which is elliptical in shape, to one that is composed of two intersecting parabolas in the principal stress plane.

Homogenization of stratified thermoviscoplastic materials

In the present paper we study the homogenization of the system of partial differential equations \[ ρ ε ( x ) ∂ v ε ∂ t − ∂ ∂ x ( μ ε ( x , θ ε ) ∂ v ε ∂ x ) = f , c ε ( x , θ ε ) ∂ θ ε ∂ t = μ ε ( x , θ ε ) ( ∂ v ε ∂ x ) 2 , \begin {array}{l} \displaystyle {\rho ^\varepsilon (x) {\partial v^\varepsilon \over \partial t} - {\partial \over \partial x} \left (\mu ^\varepsilon (x,\theta ^\varepsilon ) {\partial v^\varepsilon \over \partial x}\right ) = f,} \displaystyle {c^\varepsilon (x,\theta ^\varepsilon ) {\partial \theta ^\varepsilon \over \partial t} = \mu ^\varepsilon (x,\theta ^\varepsilon ) \left ({\partial v^\varepsilon \over \partial x}\right )^2,} \end {array} \] posed in a &gt; x &gt; b a &gt; x &gt; b , 0 &gt; t &gt; T 0 &gt; t &gt; T , completed by boundary conditions on v ε v^\varepsilon and by initial conditions on v ε v^\varepsilon and θ ε \theta ^\varepsilon . The unknowns are the velocity v ε v^\varepsilon and the temperature θ ε \theta ^\varepsilon , while the coefficients ρ ε \rho ^\varepsilon , μ ε \mu ^\varepsilon and c ε c^\varepsilon are data which are assumed to satisfy 0 &gt; c 1 ≤ μ ε ( x , s ) ≤ c 2 , 0 &gt; c 3 ≤ c ε ( x , s ) ≤ c 4 , 0 &gt; c 5 ≤ ρ ε ( x ) ≤ c 6 , − c 7 ≤ ∂ μ ε ∂ s ( x , s ) ≤ 0 , | c ε ( x , s ) − c ε ( x , s ′ ) | ≤ ω ( | s − s ′ | ) . \begin{gather*} 0 &gt; c_1 \leq \mu ^\varepsilon (x,s) \leq c_2, \quad 0 &gt; c_3 \leq c^\varepsilon (x,s) \leq c_4,\quad 0 &gt; c_5 \leq \rho ^\varepsilon (x) \leq c_6, \displaystyle {- c_7 \leq {\partial \mu ^\varepsilon \over \partial s} (x,s) \leq 0, \quad |c^\varepsilon (x,s) - c^\varepsilon (x,s’)|\leq \omega (|s - s’|).} \end{gather*} This sequence of one-dimensional systems is a model for the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and a temperature-dependent rate of plastic work converted into heat. Under the above hypotheses we prove that this system is stable by homogenization. More precisely one can extract a subsequence ε ′ \varepsilon ’ for which the velocity v ε ′ v^{\varepsilon ’} and the temperature θ ε ′ \theta ^{\varepsilon ’} converge to some homogenized velocity v 0 v^0 and some homogenized temperature θ 0 \theta ^0 which solve a system similar to the system solved by v ε v^\varepsilon and θ ε \theta ^\varepsilon , for coefficients ρ 0 \rho ^0 , μ 0 \mu ^0 and c 0 c^0 which satisfy hypotheses similar to the hypotheses satisfied by ρ ε \rho ^\varepsilon , μ ε \mu ^\varepsilon and c ε c^\varepsilon . These homogenized coefficients ρ 0 \rho ^0 , μ 0 \mu ^0 and c 0 c^0 are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient c 0 c^0 in general depends on the temperature even if the heterogeneous heat coefficients c ε c^\varepsilon do not depend on it.

Multiscale Thermodynamic Significance of the Scale Invariance Approach in Continuum Inelasticity

AbstractIn this paper, the scale invariance approach from micro- to macro-plasticity by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach,” ASME J. Eng. Mater., 117, pp. 352–355) and Ning and Aifantis (1996, “Anisotropic Yield and Plastic Flow of Polycristalline Solids,” Int. J. Plasticity, 12, pp. 1221–1240) is investigated within Rice’s normality structure (1971, “Inelastic Constitutive Relations for Solids: An Integral Variable Theory and its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455; 1975, “Continuum Mechanics and Thermodynamics of Plasticity in Relation to Microscale Deformation Mechanisms,” Constitutive Equations in Plasticity, A. S. Argon, ed., MIT Press, Cambridge, MA, pp. 23–79). The normality structure provides a minimal framework of multiscale thermodynamics, and the dissipation equivalence between the microscale and macroscale is ensured by a variational equation which can be further formulated into principle of maximum equivalent dissipation. It is revealed in this paper that within the framework of normality structure, the so-called hypothesis of generalized scale invariance holds for the kinetic rate laws, flow rules, and orthogonality conditions in the sense of Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”). Stemming from Rice’s kinetic rate laws, the generalized scale invariance reflects the inherent self-consistent character of the normality structure. If the plastic work rate is assumed to be equal to the intrinsic dissipation rate, the kinematic hardening plasticity as a demonstration of the scale invariance approach by Aifantis (1995, “From Micro- to Macro-plasticity: The Scale Invariance Approach”), can be well accommodated within the framework of normality structure. Therefore, the scale invariance approach is justified from a multiscale thermodynamic viewpoint. It is further shown that the maximization procedure in this approach just corresponds to the principle of maximum equivalent dissipation.

Elastic/plastic models for soils and sands

Some recent advances in the constitutive modelling of soils, sands and other granular materials are reviewed. These new ideas are based on the use of the concepts of the modern theory of the thermomechanics of continua. Most existing engineering theories of soil behaviour are in the nature of “recipes”, providing rules by which yield loci, flow and hardening rules and failure lines may be constructed to provide models which predict the response and failure of granular materials in a limited set of laboratory experiments. These models rarely have any firm physical basis. The thermomechanical approach, on the other hand, enables us to construct models, which both satisfy the fundamental laws of thermodynamics and have definite physical interpretations in terms of energy storage and dissipation. Of particular importance is the distinction between the rate of plastic work and plastic dissipation, the intimate connection between dissipation and the yield loci and flow rule, and between plastic dilation and induced anisotropy. These ideas lead naturally to the concepts of critical state soil mechanics. In addition the thermomechanical approach offers the real possibility of relating the observed and simulated behaviour of granular materials on the micro- and meso-scales, to the observed behaviour at the continuum level. Some initial steps in this direction are outlined at the end of this paper.

On the relationship between stress–dilatancy, anisotropy, and plastic dissipation for granular materials

Techniques of modern thermomechanics are used to analyse the stress–dilatancy relation for shear deformations of frictional, granular materials. Central to this approach is the recognition that in general, deformations of granular materials can involve stored plastic work. It is recognised that in shear deformations, where the shape of the grains induces the dilatancy, the confining pressure is a constraint and dissipates no energy. Nevertheless it is shown that, in such a shear deformation, the total plastic work rate is equal to the rate of energy dissipation, and that this is given by Thurairajah's classical observations. It is further shown that anisotropy is necessarily induced, and that the stress–dilatancy relation is given by Taylor's well-known formula. It is also demonstrated that the extant procedure for deducing yield conditions and flow rules from specified dissipation functions is invalid.

Heat Transfer in Friction Stir Welding—Experimental and Numerical Studies

In the friction stir welding (FSW) process, heat is generated by friction between the tool and the workpiece. This heat flows into the workpiece as well as the tool. The amount of heat conducted into the workpiece determines the quality of the weld, residual stress and distortion of the workpiece. The amount of the heat that flows to the tool dictates the life of the tool and the capability of the tool for the joining process. In this paper, we formulate the heat transfer of the FSW process into two boundary value problems (BVP)—a steady state BVP for the tool and a transient BVP for the workpiece. To quantify the physical values of the process the temperatures in the workpiece and the tool are measured during FSW. Using the measured transient temperature fields finite element numerical analyses were performed to determine the heat flux generated from the friction to the workpiece and the tool. Detailed temperature distributions in the workpiece and the tool are presented. Discussions relative to the FSW process are then given. In particular, the results show that (1) the majority of the heat generated from the friction, i.e., about 95%, is transferred into the workpiece and only 5% flows into the tool and (2) the fraction of the rate of plastic work dissipated as heat is about 80%.

Role of plasticity in heat generation during high rate deformation and fracture of polycarbonate

Abstract The role of plastic deformation in generating heat during dynamic compression and fracture of polycarbonate was examined in a combined experimental and numerical investigation. Split Hopkinson pressure bar (SHPB) experiments and opening mode dynamic fracture experiments were performed to measure the thermomechanical response of polycarbonate at various loading conditions. A companion set of dynamic, thermomechanically coupled explicit finite element simulations of the tested geometries was performed to isolate the temperature increase of the polymer due to plasticity effects only, and these were then compared to the experiments to identify the role of plasticity in the overall heating of the polymer. Temperature measurements during the experiments were made with a remote sensing technique that utilizes the detection of infrared radiation. Polycarbonate was observed to have a rate dependent yield and plastic deformation response under uniaxial compression. For the strain rates tested in the SHPB experiments (400 to 3000 s−1), the fraction of plastic work rate converted to heat was found to be approximately 0.5. The dynamic fracture experiments indicated that the maximum temperature increase in the region surrounding the propagating crack was over 100 K, and the crack tip velocity was limited to approximately 60% of the Rayleigh wave speed. The numerical simulations used the plastic behavior of polycarbonate observed during the compression SHPB experiments as inputs for an incremental plasticity model that uses a rate dependent Mises yield surface and isotropic hardening. Plastic deformation was the only source of material heating in the simulations. Good agreement between the SHPB experiments and the numerical simulations was seen for both stress and temperature during the compression event. Crack growth during the dynamic fracture simulations was imposed by a manual node release technique that duplicated the crack growth speed observed in the experiments. The dynamic fracture simulations which only included the thermoplastic effect significantly under-predicted the temperature increase in the region surrounding the propagating crack. The results indicate that plastic deformation is not the dominating source of heat generation during the dynamic fracture of polycarbonate, it only accounts for about 8% of the measured heating. Additional effects, such as thermofracture coupling, must be considered in the simulations if a more accurate comparison is to be made with experiments.

A new look at energy release rate in fracture mechanics

The energy balance for fracture in elastic/perfectly plastic solids is examined using the finite element method. An extension-release procedure that gives numerically converged solutions is employed in the numerical simulation of crack extensions in elastic/plastic solids. Increments of work and energy during crack extension are calculated for various loading conditions. Several conclusions are obtained. First, the elastic separation work of creating new crack surfaces is shown to be negligible, indicating that the Griffith-type energy release does not exist. Second, as the yield stress increases, the plastic dissipation work rate associated with crack extension converges to the energy release rate in the limiting elastic solid. The latter result can be adopted to interpret the classical energy release rate in elastic solids as plastic dissipation work rate taken in the limit as the yield stress approaches infinity during crack extension. Lastly, it is shown that the energy release rate obtained according to Irwin's plastic zone adjustment approach is equal to the plastic dissipation work rate for the original crack, provided the plastic zone size is less than 10% of the original crack size.

A shear lip analysis of concave R -curves for an AlMgZn alloy

A concave R-curve for an AlMgZn alloy covers the transition from fully flat to fully slant fracture in 25 mm thick specimens of varying geometry. The transition is associated with the onset of plane stress conditions near the crack front. The concave R-curve is analysed using the increasing width of the shear lips, as in the Krafft et al. model, which increased the plastic work parabolically. The elastic and plastic work rates are evaluated using parabolic curve fits to the R-curve data. The present study showed that the plastic work rate scales with the crack extension from fully flat to fully slant fracture, which is geometry dependent