Plastic Work Rate Research Articles

A finite deformation formulation for amorphous glassy polymers under moderate and impact strain rates: Application to adhesive films

This work describes a novel finite strain hypo-elastic formulation for amorphous thermoset polymers working in the glassy regime. The constitutive formulation is able to describe time and pressure-dependent behaviour under loading rates ranging from quasi-static to impact-type loading conditions and to account for thermo-mechanical coupling effects. A non-linear pressure dependent potential function is introduced to capture the non-associative visco-plastic potential flow for volumetric plastic strain control. The formulation also incorporates an internal state or deformation resistance variable to enable the description of the polymer hardening — softening behaviour, as well as a novel relation for the dissipative fraction of the plastic work in terms of the equivalent accumulated plastic strain. The constitutive model is formulated within a finite strain kinematics framework, and full details of its numerical implementation into the finite element method using a fully implicit Euler Backward algorithm are given. Model calibration is carried out for three different thermosetting resins (PR520 and RTM6 Epoxies, and MMA adhesives).To illustrate the model capabilities, two case studies are investigated: (i) the de-formation behaviour of epoxy under moderate and impact-type loading conditions under isothermal and fully adiabatic conditions, and (ii) the fracture behaviour of adhesive layers used to bond stiff metallic substrates. Results on RTM6 epoxy show that at, e.g., 60% true strain, 48.5% of the rate of plastic work is dissipative, and that the corresponding predicted increase in temperature due to the locally dissipated heat is consistent with published calorimetry data on a similar thermoset polymer. It was also found that, by coupling the proposed constitutive model with a cohesive zone model, it was possible to predict accurately the effect of an MMA adhesive layer thickness and pressure-dependency on the growth of an interfacial crack between the MMA adhesive and the metallic substrates. The proposed model should constitute a generic phenomenological formulation for the mechanical behaviour prediction of a wide range of thermoset polymers under confined and quasi-adiabatic conditions such as in composites and adhesive joints.

In Situ Prediction of Metal Fatigue Life Using Frequency Change

A reliable technique for rapid prediction of the remaining useful life (RUL) of components experiencing fatigue degradation is introduced. The approach is based on measuring the temperature signature of a component upon rapidly changing its operating frequency for a short period of time. The temperature variations caused by alterations in plastic work rate are correlated to the loading history. The efficacy of the approach is investigated by conducting a series of axial fatigue tests on stainless steel 316 specimens. The material characterization involves subjecting the material to a constant amplitude fatigue load at 4 Hz and 12 Hz frequencies. The operating frequency is temporarily adjusted to the characterization frequencies for a brief duration. During this period, the change in the slope of temperature rise is recorded. Subsequently, the operation frequency is reverted to its original state, and the remaining useful life is predicted based on the recorded data. The model provides predictions for operation frequencies of 6 Hz, 8 Hz, and 12 Hz, and notably, the error of predictions is consistently under 12% for all cases. The method allows the operator to reliably estimate the remaining usefulness for field applications without interrupting the operation.

Thermodynamically consistent strain hardening variable/driving force, inelastic stored energy and self-heating in dynamic plasticity

Accounting for self-heating and subsequent thermal softening in dynamic plasticity requires an accurate estimation of the mechanical dissipation rate which is by definition the difference between the plastic work rate and the inelastic stored energy rate. Yet, in constitutive modeling, the evaluation of the inelastic stored energy strongly depends on the adopted approach, viz. irreversible thermodynamics vs microstructure-motivated approach, and, whatever the approach, does not yet lead to realistic results in terms of self-heating. The aim of the present study is to attempt to conciliate both above mentioned approaches by building constitutive models that relate strain hardening, inelastic stored energy and self-heating in a consistent way. The models in question are developed in agreement with experimental results obtained on five high-strength steels.

An analytical model for the prediction of rolling force of thin slab

ABSTRACTA rolling force model is established by using the unified yield criterion, in which both the changes of the slab temperature and the radius of the roller are considered. In order to solve the integral problem of the nonlinear specific plastic work rate, the unified yield criterion is used to analyse a simplified velocity field, and the internal deformation power is derived. Then, the shear power and friction power are also calculated based on the velocity field, and the total power is obtained by summarising these derived formulas. Finally, the analytic solution of rolling force in terms of the unified yield criterion is obtained by the variational method. Thereafter, the measured data from one domestic factory is used to check the analytical solution of rolling force, in which the temperature drop from the slab inlet to its outlet is considered. Meanwhile, the roller flattening is also considered. It is shown from the comparison and discussion that: the Mises yield criterion that is a special case of the unified yield criterion is more accurate in predicting the rolling force; the thinner the slab is, the more the temperature drop is; the calculation results by considering the roll flattening are improved.

An Experimental Investigation of the Mechanical Behavior and Particle Crushing Characteristic of Volcanic Soil.

Volcanic soil is a special soil that is well-known for its distinctive texture, vesicular nature, and particle fragility. The fragility characteristic of volcanic soil is the main factor affecting the foundation stability in road engineering. This study focuses on the mechanical properties and particle crushing characteristics of volcanic soil retrieved from Northeast China. A series of triaxial consolidation and drainage shear tests are performed on volcanic coarse-grained soil (5 mm > d > 0.075 mm) under different initial relative densities and effective confining pressures. Results show the peak friction angle of volcanic soil significantly decreases with the increase of confining pressure. The particle crushing degree of volcanic soil increases with the increase of confining pressure, particle size, and relative density. The relative breakage rate of the same particle size group has a good linear relationship with a fractal dimension. Moreover, for the same particle size, the relationship between plastic work and relative breakage rate can be fitted by a power function, which is not significantly affected by relative density or effective confining pressure. From an engineering view, in addition to increasing the compaction degree of volcanic soil, volcanic soil with fine particles used as a roadbed filler can significantly reduce the deformation of the roadbed and improve the bearing capacity of the foundation.

A constitutive model for poly-methyl-methacrylate over a wide range of strain rates and temperatures

This study proposes a three-dimensional elasto-viscoplastic constitutive model to depict the rate- and temperature-dependent behaviour of poly-methyl-methacrylate (PMMA) under a variety of strain rates and temperature ranges at finite deformations. Yield stress is described via incorporating an appropriate Young's modulus into the Argon's model. The effects of temperature and strain rate are incorporated into the Young's modulus. Subsequently, a new empirical equation has been suggested to describe the softening behaviour over a wide range of strain rates and temperatures. Further, the temperature dependence of the fraction of plastic work rate converted to heat (β factor) has been studied. In addition, the expressions of hardening parameters (Cr and N) under a changing temperature in the full network model that represents the strain-hardening characteristics of polymers have been proposed. Moreover, a method to calibrate the values of newly introduced parameters has been introduced as well. The experimental results demonstrate that the new softening model can predict softening behaviour at various strain rates (from 0.0003 s−1 to 4300 s−1) and temperatures (from 298 K to the glass transition temperature θg).

Particle breakage characteristics of a foundation filling material on island-reefs in the South China Sea

Calcareous sand is the only building material in the reclaimed foundation of island-reefs in the South China Sea, and calcareous sand in reclaimed foundation is predominantly calcareous gravelly sand (CGS). Particle breakage is the primary factor affecting the stability of reclaimed foundation. Considering the susceptibility to breakage of calcareous sand, multiple triaxial shear tests are conducted to investigate the particle breakage and evolution of CGS under different relative densities and effective confining pressures. Test results show that relative breakage rate more satisfactorily reflects the overall particle breakage degree of sample rather than grading parameters and fractal dimension. Thus, the relationship of relative breakage rate with relative density and effective confining pressure is established. A trend of “contraction first and dilatancy afterwards” is observed for CGS, which makes plastic work done by mean effective stress exist the negative value. Besides, a plastic work calculation model is developed to reflect the effects of relative density and effective confining pressure. Moreover, the relationship between plastic work and relative breakage rate can be fit using an inverse proportional function and is not significantly affected by relative density or effective confining pressure. These findings in this study lay a theoretical foundation for assessing the stability of reclaimed foundation of island-reefs.

Mesoscale Mechanisms in Viscoplastic Deformation of Metals and Their Applications to Constitutive Models.

Deformation of metals has attracted great interest for a long time. However, the constitutive models for viscoplastic deformation at high strain rates are still under intensive development, and more physical mechanisms are expected to be involved. In this work, we employ the newly-proposed methodology of mesoscience to identify the mechanisms governing the mesoscale complexity of collective dislocations, and then apply them to improving constitutive models. Through analyzing the competing effects of various processes on the mesoscale behavior, we have recognized two competing mechanisms governing the mesoscale complex behavior of dislocations, i.e., maximization of the rate of plastic work, and minimization of the elastic energy. Relevant understandings have also been discussed. Extremal expressions have been proposed for these two mesoscale mechanisms, respectively, and a stability condition for mesoscale structures has been established through a recently-proposed mathematical technique, considering the compromise between the two competing mechanisms. Such a stability condition, as an additional constraint, has been employed subsequently to close a two-phase model mimicking the practical dislocation cells, and thus to take into account the heterogeneous distributions of dislocations. This scheme has been exemplified in three increasingly complicated constitutive models, and improves the agreements of their results with experimental ones.

An Exact Axisymmetric Solution in Anisotropic Plasticity

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed.

Bifurcation theory of plasticity, damage and failure

I developed a comprehensive theory of plasticity, damage and failure, which is independent of the micromechanical mechanism of plasticity. It includes all major components of the mechanical behavior of ductile materials, e.g. strain/work-hardening and Bauschinger effect, and covers all regimes of viscoplastic tensile/compressive loading/unloading. The theory is based on the concepts of free energy and damage parameter and has the attributes of both rate-dependent and rate-independent plasticity, kinematic and isotropic hardening. The ultimate failure of a specimen is defined as a point of critical value of the damage parameter. The theory provides justification for Drucker’s ad hoc condition of instability. The theory describes diverse phenomena of Lüders stress, creep, internal friction, and fatigue on the same self-consistent basis. In quasi-static low-cycle fatigue, I derive a Paris-type equation between the rate of degradation and cyclic plastic work and reveal a Coffin-Manson power-law relationship between the number of cycles to failure and plastic-strain amplitude. In dynamic fatigue, I find two regimes—low-cycle and high-cycle—whose loading amplitudes are separated by the yield point. The two regimes have significantly different dependencies on the frequency and viscosity. In this publication, I present the theory and computer simulations of a homogeneous viscoplastic body subjected to various conditions of isothermal strain-controlled uniaxial loading. In the follow-up publications, I will expand the theory on the cases of inhomogeneous three-dimensional bodies subjected to multiaxial loading at variable temperatures and compare the results to the experiments. Complete version of the theory can be used for the full-cycle cradle-to-grave design of new materials in service.

Proposal and application of a new yield criterion for metal plastic deformation

A new linear yield criterion in terms of principal stresses is developed for the purpose of overcoming the analytical difficulty of mechanical parameters when using the nonlinear Mises yield criterion. This developed yield criterion is established through the parabola interpolation method and the mathematical averaging method, which can be called as the mean influence factor yield criterion, or called MIF yield criterion for short. The yield locus of the criterion on the $$\pi $$-plane is an equilateral and non-equiangular dodecagon, lies between the Tresca and TSS loci, and is very close to the locus of the Mises yield criterion. By comparing with the classical yield criteria, the MIF yield criterion varies linearly with the Lode parameter and has a good accuracy with experimental data. Moreover, based on the flow rule and the geometric projection of the principle stress components, the specific plastic work rate of the yield criterion is also derived. The criterion and its specific plastic work rate are used to solve the limit load of a simply supported circular plate and the forging of a rectangular bar, and both the corresponding analytical solutions are obtained. The theoretical results for the limit load and the forging force are in good agreement with the simulation and experimental data, respectively, and are more accurate than the results calculated from other yield criteria.

A weighted average yield criterion and its applications to burst failure of pipeline and three-dimensional forging

By weighting the Tresca and TSS yield criteria with the golden ratio 0.618, a linear yield criterion, called WA yield criterion for short, is proposed as a linear function of the principal stress components σ1, σ2, and σ3. The shear yield stress for the proposed yield criterion is also calculated, whose value is very close to that of the Mises yield criterion. The yield locus of the proposed yield criterion on the π-plane is described in geometry and proved to be a dodecagon with the same side length of 0.4148σs and two intersection angles of 140.4° and 159.6°. To verify the proposed yield criterion, a comparison is made between the calculated stress difference ratio and the available experimental data, and a good agreement is found. By applying the WA yield criterion to the prediction of burst pressure for pipelines, the present yield criterion is found to be the best one. Moreover, the specific plastic work rate (SPWR) of the WA yield criterion is also derived and applied to a three-dimensional forging accounting for bulge. Through the derived SPWR, an analytical solution of the forging forces is obtained, and an experiment of forging pure lead is performed. Comparison of the present calculated forging forces with the present experimental data of pure lead forging as well as the other theoretical ones clarifies that the present SPWR has the superiority of predicting the forging forces. In addition, the dominant parameters that affect the burst pressure of pipelines and the stress state coefficients of the 3D forging are discussed systematically.

On the homogeneous nucleation and propagation of dislocations under shock compression

The dynamic response of crystalline materials subjected to extreme shock compression is not well understood. The interaction between the propagating shock wave and the material’s defect occurs at the sub-nanosecond timescale which makes in situ experimental measurements very challenging. Therefore, computer simulation coupled with theoretical modelling and available experimental data is useful to determine the underlying physics behind shock-induced plasticity. In this work, multiscale dislocation dynamics plasticity (MDDP) calculations are carried out to simulate the mechanical response of copper reported at ultra-high strain rates shock loading. We compare the value of threshold stress for homogeneous nucleation obtained from elastodynamic solution and standard nucleation theory with MDDP predictions for copper single crystals oriented in the [0 0 1]. MDDP homogeneous nucleation simulations are then carried out to investigate several aspects of shock-induced deformation such as; stress profile characteristics, plastic relaxation, dislocation microstructure evolution and temperature rise behind the wave front. The computation results show that the stresses exhibit an elastic overshoot followed by rapid relaxation such that the 1D state of strain is transformed into a 3D state of strain due to plastic flow. We demonstrate that MDDP computations of the dislocation density, peak pressure, dynamics yielding and flow stress are in good agreement with recent experimental findings and compare well with the predictions of several dislocation-based continuum models. MDDP-based models for dislocation density evolution, saturation dislocation density, temperature rise due to plastic work and strain rate hardening are proposed. Additionally, we demonstrated using MDDP computations along with recent experimental reports the breakdown of the fourth power law of Swegle and Grady in the homogeneous nucleation regime.

A general kinematically admissible velocity field for axisymmetric forging and its application to hollow disk forging

An approach for selecting general kinematically admissible velocity fields for axisymmetric forging problems is outlined. The approach accounts for the existence of a rigid zone at frictional interfaces and for the singular behavior of real velocity fields in the vicinity of maximum friction surfaces. The plastic work rate for a material obeying the von Mises yield criterion and its associated flow rule is expressed in terms of one arbitrary function of a single argument, its derivative, and anti-derivative. An upper bound solution for constrained forging is given. Comparison with an available upper bound solution is made and it is shown that the new kinematically admissible velocity field results in a more accurate solution at high friction. Other problems that can be treated are briefly discussed.

Effect of the yield criterion on the strain rate and plastic work rate intensity factors in axisymmetric flow

The main objective of the present paper is to study the effect of the yield criterion on the magnitude of the strain rate and plastic work rate intensity factors in axisymmetric flow of isotropic incompressible rigid perfectly plastic material by means of a problem permitting a closed-form solution. The boundary value problem consisting of the axisymmetric deformation of a plastic tube is solved. The outer surface of the tube contracts. The radius of the inner surface does not change. The material of the tube obeys quite a general yield criterion and its associated flow rule. The maximum friction law is assumed at the inner surface of the tube. Therefore, the velocity field is singular near this surface. In particular, the strain rate and plastic work rate intensity factors are derived from the solution. It is shown that the strain rate intensity factor does not depend on the yield criterion but the plastic work rate intensity factor does.

Average apothem yield criterion and its application to strip forging

A new linear yield criterion, referred to as average apothem (AA) yield criterion is developed in this paper based on the Tresca and Twin Shear Stress (TSS) yield criteria. This criterion is a piecewise linear function and its locus on the π-plane in Haigh Westergaard stress space is an equilateral and non-equiangular dodecagon. Comparison with the Tresca and Mises yield criteria shows that the locus of the proposed yield criterion lies between those of the Tresca and Mises loci and is much closer to that of Mises yield criterion. Further, its specific plastic work rate is developed. As an application of the proposed criterion, the AA yield criterion is applied to the analysis of a generalized worked example of strip forging and an analytical solution of the stress effective factor is obtained. Comparison with the results by Tresca, Mises and TSS yield criteria shows that the proposed criterion provides an intermediate solution between the results of Tresca and Mises criteria.

Heat generation in plane strain compression of a thin rigid plastic layer

This paper presents a theoretical investigation into heat generation in the continued quasi-static plane strain compression of a thin metal strip between two rigid, parallel perfectly rough dies. The strip material is rigid perfectly plastic. The length of the dies is supposed to be much larger than the current strip thickness. The plastic work rate approaches infinity in the vicinity of perfectly rough friction surfaces. Since the plastic work rate is involved in the heat conduction equation, this significantly adds to the difficulties of solutions of this equation. In particular, commercial finite element packages are not capable of solving such boundary value problems. The present approximate solution is given in Lagrangian coordinates. In this case, the original initial/boundary value problem reduces to the standard second initial/boundary value problem for the nonhomogeneous heat conduction equation. Therefore, the Green’s function is available in the literature. An example is presented to illustrate the general solution.

On the recoverable and dissipative parts of higher order stresses in strain gradient plasticity

The expressions for the free energy in two recent formulations of strain gradient plasticity are extended to include the locked-in strain energy around statistically stored dislocations. This is accomplished by using the strain dependent factor η(ep), which represents the fraction of the rate of plastic work converted into heat in accordance with the latent heat measurements from classical metal plasticity. The expressions for the plastic work in the two formulations differ by different representations of the portion of plastic work associated with the existence of plastic strain gradients and the corresponding network of geometrically necessary dislocations, while the dissipative parts of plastic work are assumed to be the same in both formulations. The expressions for the recoverable and dissipative parts of the higher order stresses, defined as the work-conjugates to plastic strain and its gradient, are then derived. It is shown that the stress and strain fields of isothermal boundary-value problems of strain gradient plasticity are independent of η, but that this factor may be of importance for non-isothermal analysis in which the dissipated plastic work acts as an internal heat source. The effects of plastic strain gradient on the plastic response of twisted hollow circular tubes made of a rigid-plastic material with different hardening properties are then evaluated and discussed.

Equal perimeter yield criterion and its specific plastic work rate: Development, validation and application

In order to overcome the nonlinearity of Mises criterion, a new linear yield criterion with a dodecagon shape of the same perimeter as Mises criterion was derived by means of geometrical analysis. Its specific plastic work rate expressed as a linear function of the yield stress, the maximum and minimum strains was also deduced and compared with that of Mises criterion. The physical meaning of the proposed yield criterion is that yielding of materials begins when the shear yield stress τs reaches the magnitude of 0.594σs. By introducing the Lode parameter, validation of evolution expressions of the proposed yield criterion with those based on Tresca, Mises and TSS criteria as well as available classical yield experimental results of various metals shows that the present results intersect with Mises results and coincide well with experimental data. Moreover, further application to the limit analysis of circle plate as an example is performed to demonstrate the effectiveness of the proposed yield criterion, and the subsequent comparison of limit loads with the Tresca analytical solutions and Mises numerical results shows that the present results are higher than the Tresca analytical results, and are in good agreement with the Mises numerical results.

The strain rate intensity factor in the plane strain compression of thin anisotropic metal strip

Using an available analytic solution for instantaneous plane strain compression of a plastically anisotropic strip between two parallel plates the strain rate intensity factor is found assuming Hill’s quadratic yield criterion. The distribution of material properties is uniform. The effect of parameters characterizing plastic anisotropy of the strip on the magnitude of the strain rate intensity factor is demonstrated. A possibility to replace the strain rate intensity factor with the plastic work rate intensity factor is discussed. Singular behavior of the plastic spin in the vicinity of the friction surface is revealed and discussed.