The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity

A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J ₂ flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.     

Time-dependent behavior of passive skeletal muscle

An isotropic three-dimensional nonlinear viscoelastic model is developed to simulate the time-dependent behavior of passive skeletal muscle. The development of the model is stimulated by experimental data that characterize the response during simple uniaxial stress cyclic loading and unloading. Of particular interest is the rate-dependent response, the recovery of muscle properties from the preconditioned to the unconditioned state and stress relaxation at constant stretch during loading and unloading. The model considers the material to be a composite of a nonlinear hyperelastic component in parallel with a nonlinear dissipative component. The strain energy and the corresponding stress measures are separated additively into hyperelastic and dissipative parts. In contrast to standard nonlinear inelastic models, here the dissipative component is modeled using an evolution equation that combines rate-independent and rate-dependent responses smoothly with no finite elastic range. Large deformation evolution equations for the distortional deformations in the elastic and in the dissipative component are presented. A robust, strongly objective numerical integration algorithm is used to model rate-dependent and rate-independent inelastic responses. The constitutive formulation is specialized to simulate the experimental data. The nonlinear viscoelastic model accurately represents the time-dependent passive response of skeletal muscle.     

Dissipative flow model based on dissipative surface and irreversible thermodynamics

Summary A dissipative flow model is presented to describe dissipative deformation processes in a macroscopic solid continuum. Dissipative process may consist of material plasticity, material damage and other intrinsic mechanical phenomena represented by internal variables. The concept of a dissipative surface is introduced in the paper to distinguish between conservative and dissipative processes. Conventional plastic yielding and damage initiation are expressed by a unique condition which may else include other possible intrinsic mechanical dissipations. The proposed model is based on the principles of irreversible thermodynamics and the minimum free energy theorem. A modified material stability postulate, modified Drucker's postulate, in thermodynamic stress space is also used to obtain the same results. Received 1 July 1998; accepted for publication 13 January 1999     

Determination of the Constitutive Relation Parameters of a Metallic Material by Measurement of Temperature Increment in Compressive Dynamic Tests

The mechanical behaviour of a material can be established by an analytic expression called the constitutive relation that shows stress as a function of plastic strain, strain rate, temperature, and possibly other thermo-mechanical variables. The constitutive relation usually includes such parameters as coefficients or exponents that must be determined. At a high strain rate, the heat generated during the deformation process is directly related to the plastic deformation energy of the material. This energy can be calculated from the plastic work, resulting in an expression that includes the constitutive relation parameters as variables. The heat generated can also be estimated by measuring the temperature surface of the specimen during compressive tests using the technique of infrared thermography. The objective of this paper is to present a procedure for determining the constitutive relation parameters by measuring the temperature increase associated with plastic strain in compressive Hopkinson tests. The procedure was applied to estimate the parameters of the Johnson–Cook constitutive relation of an aluminium alloy (Al6082).     

Plastic instability studied experimentally on a semi-crystalline polymer through thermomechanical heat source identification: the flow stress concept revisited

Micromechanical deformation phenomena such as those leading to macroscopic viscoelastic and plastic behavior must be studied from a thermodynamic viewpoint, as they induce complex and partly irreversible heat effects. Calorimetric measurements of the intrinsic volumetric thermomechanical heat sources (THS) activated in the material bulk during mechanical loads can produce valuable information with respect to that aim. They can be based on infrared imaging if submitted to inverse algorithms that allow a correct reconstruction of THS to be produced. Here, an inverse method relying on a diffusion-advection heat transfer model is applied to experimental temperature maps recorded during tensile tests. These are made on a semi-crystalline polymer that shows a strong development of plastic instabilities. Along with simultaneous kinematic observables produced with a digital image correlation system, the competition between advection and diffusion phenomena may be clearly established. 1-D profiles of the reconstructed THS and measured strain rates illustrate clearly that thermomechanical effects associated with necking onset and propagation follow the kinematic variable in a rather direct manner. Finally, we show for tensile experiments that THS estimations lead to analyze plasticity as a rheological behavior controlled by the flow stress, responsible of necking development and propagation.     

The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 2: single-crystal plasticity

Variational formulations are constructed for rate-independent problems in small-deformation single-crystal strain-gradient plasticity. The framework, based on that of Gurtin (J Mech Phys Solids 50: 5–32, 2002), makes use of the flow rule expressed in terms of the dissipation function. Provision is made for energetic and dissipative microstresses. Both recoverable and non-recoverable defect energies are incorporated into the variational framework. The recoverable energies include those that depend smoothly on the slip gradients, the Burgers tensor, or on the dislocation densities (Gurtin et al. J Mech Phys Solids 55:1853–1878, 2007), as well as an energy proposed by Ohno and Okumura (J Mech Phys Solids 55:1879–1898, 2007), which leads to excellent agreement with experimental results, and which is positively homogeneous and therefore not differentiable at zero slip gradient. Furthermore, the variational formulation accommodates a non-recoverable energy due to Ohno et al. (Int J Mod Phys B 22:5937–5942, 2008), which is also positively homogeneous, and a function of the accumulated dislocation density. Conditions for the existence and uniqueness of solutions are established for the various examples of defect energy, with or without the presence of hardening or slip resistance.     

Model for charge/discharge-rate-dependent plastic flow in amorphous battery materials

Plastic flow is an important mechanism for relaxing stresses that develop due to swelling/shrinkage during charging/discharging of battery materials. Amorphous high-storage-capacity Li–Si has lower flow stresses than crystalline materials but there is evidence that the plastic flow stress depends on the conditions of charging and discharging, indicating important non-equilibrium aspects to the flow behavior. Here, a mechanistically-based constitutive model for rate-dependent plastic flow in amorphous materials, such as Li_xSi alloys, during charging and discharging is developed based on two physical concepts: (i) excess energy is stored in the material during electrochemical charging and discharging due to the inability of the amorphous material to fully relax during the charging/discharging process and (ii) this excess energy reduces the barriers for plastic flow processes and thus reduces the applied stresses necessary to cause plastic flow. The plastic flow stress is thus a competition between the time scales of charging/discharging and the time scales of glassy relaxation. The two concepts, as well as other aspects of the model, are validated using molecular simulations on a model Li–Si system. The model is applied to examine the plastic flow behavior of typical specimen geometries due to combined charging/discharging and stress history, and the results generally rationalize experimental observations.     

A differential model for the rheological properties of concentrated suspensions with weakly viscoelastic matrices

A model for the rheological properties of a concentrated suspension in weakly viscoelastic fluid matrices is proposed. The model is derived according to the Roscoe differential procedure described in 1952. The analytical results produced recently by Greco et al. (J Non-Newton Fluid Mech 147:1–10, 2007) and Housiadas and Tanner (J Non-Newton Fluid Mech 162:88–92, 2009) for dilute suspensions of neutrally buoyant, non-Brownian rigid spheres in weakly viscoelastic matrix fluids are the key results which are used as a base to predict the properties of concentrated suspensions. The results are compared with the few available experimental data from the literature, showing promising trends for the viscometric properties of the suspensions. In particular, one sees the rapidly increasing value of −N₂/N₁ as concentration increases.     

A principle of virtual work for combined electrostatic and mechanical loading of materials

The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.     

Use of structural components of specific work of internal forces for estimating the strength of viscoelastic structures in local stress concentration regions

Material fracture experiments on specimens and structures testify that materials can resist greater stresses in local stress concentration regions than in regions with a nearly homogeneous stress state. Taking this fact into account in design stress analysis permits one to reveal additional structure loading and/or service life margins. One approach aimed at taking into account the increased strength in local stress concentration regions is to use averaged limit characteristics parametrically depending on the characteristic size L of the averaging region. One version of this approach is the concept of “elementary block” of a material [1, 2]. The averaged limit characteristics are determined by an experiment-calculation method involving the analysis of the stress-strain state of a material specimen with a stress concentrator at the time when the specimen attains the limit state preceding macrofracture.In [3], the dependence of the averaged limit separation stresses on the size of the averaging region was determined on the basis of numerical analysis of the singular stress state of the specimen used to determine the standard characteristics of the adhesion strength of a filled polymer material. In the present paper, we generalize the above approach to the case of a viscoelastic material. For the limit characteristics of the material in the local stress concentration region we take the volume-averaged components of the specific work of internal forces [4, 5] (the averaged specific absorbed energy and the averaged specific instantaneously reversible energy). The introduction of two limit energies originates from the fact that, to initiate the process of macrofracture, it is necessary to satisfy the following two conditions simultaneously: the material must be “damaged” sufficiently strongly by the preceding loading, and the “damaged” material must be loaded sufficiently strongly. As an example of determining the material averaged limit energy characteristics in a local stress concentration region, we consider the problem about the strain of a viscoelastic specimen used to determine the standard adhesion strength characteristics. The problem is solved numerically under the following assumptions: the specimen material is assumed to be linearly viscoelastic, and the specific absorbed energy in the stress concentration region is assumed to coincide in magnitude with the specific scattered energy. To estimate the accuracy of the numerical method, we use the solution of the model problem about the action of a plane circular die on a half-space consisting of a linearly viscoelastic incompressible material.     

A theoretical investigation on the vibrational characteristics and torsional dynamic response of circumferentially cracked turbo-generator shafts

Turbo-generator shafts are often subjected to complex dynamic torsional loadings, resulting in generation and propagation of circumferential cracks. Mode III fatigue crack growth generally results in a fracture surface consisting of peaks and valleys, resembling a factory roof. The fracture surface roughness depends on the material microstructure, the material yield strength, and the applied cyclic torque amplitude. This crack pattern can severely affect the vibration characteristics of the shafts. The accurate evaluation of the torsional dynamic response of the turbo-generator shafts entails considering the local sources of energy loss in the crack vicinity. The two most common sources of the energy loss are the local energy loss due to the plasticity at the crack tip and frictional energy loss due to interaction of mutual crack surfaces. A theoretical procedure for evaluating the values of the system loss factors corresponding to these sources of energy loss is presented. Furthermore, the local flexibility is obtained by evaluating the resistance of the cracked section of the shaft to the rotational displacement. The shaft material is assumed to be elastic perfectly plastic. The effects of the applied Mode III stress intensity factor and the crack surface pattern parameters on the energy loss due to the friction and the energy loss due to the plasticity at the crack tip are investigated. The results show that depending on the amplitude of the applied Mode III stress intensity factor, one of these energy losses may dominate the total energy loss in the circumferentially cracked shaft. The results further indicate that the torsional dynamic response of the turbo-generator shaft is significantly affected by considering these two sources of the local energy loss.     

Elongational viscosity of LDPE with various structures: employing a new evolution equation in MSF theory

Molecular stress function theory with new strain energy function is used to analyze transient extensional viscosity data of seven low-density polyethylene (LDPE) melts with various molecular structures as published by Stadler et al. (Rheol Acta 48:479–490, 2009) Pivokonsky et al. (J Non Newton Fluid Mech 135:58–67, 2006) and Wagner et al. (J Rheol 47(3):779–793, 2003). The new strain energy function has three nonlinear viscoelastic material parameters and assumes that the total stored energy of a branched molecule is given by different backbone and side chains stretching. The model parameters have been fitted for each LDPE in order to correlate with the supposed macromolecular structure expected from the type of synthesis. Most probable molecular structures for these LDPEs are comb and Cayley tree structures for respectively low- and high-molecular weight parts.     

A Shear Compression Disk Specimen with Controlled Stress Triaxiality under Quasi-Static Loading

This paper introduces a double shear axisymmetric specimen (Shear Compression Disk) and the methodology to extract flow and fracture properties of ductile materials, under various stress triaxiality levels. A thorough numerical investigation of the experimental set-up is performed, which reveals that the stresses are quite uniformly distributed in the gauge section during all the stages of the test. The attainable level of stress triaxiality (with pressures of up to 1.9 GPa) ranges from −0.1 to 1, which can be adjusted by a proper choice of geometrical parameters of the specimen. The methodology is implemented to quasi-static experiments on 4340 Steel and Aluminum 7075-T651 specimens. The flow properties are compared to those obtained by upsetting cylinders and show a very good agreement. For these materials it is observed that, contrary to the fracture strain, the flow properties are quite insensitive to the level of stress triaxiality. The fracture strain of the aluminum alloy increases with triaxiality and may be fitted with an exponential polynomial of the type suggested by [27]. These examples demonstrate the potential of the new specimen to obtain flow and fracture properties of ductile materials under controlled triaxiality.     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

Experimental Research on Failure Waves in Soda-Lime Glass

A series of plate impact experiment with soda-lime glass specimens in different thicknesses are conducted on a 57 mm diameter one-stage gas gun in order to further investigate the so-called failure wave phenomena under dynamic compressive loads. With the aid of the VISAR technique, the failure wave trajectory is explored, which shows that, apart from a constant failure wave velocity, an initial delay time for the failure wave to initiate at the impact surface of the specimen should be taken into consideration. Comparing our experimental results with the available data presented in the previous open literature shows that, with the increasing magnitude of the impact loads, the failure wave velocity increases and the initial delay time decreases. Moreover, the derived initial delay time τ = 0.694 μs for the soda-lime glass specimens under the impact stress of 4.7 GPa is the same order of magnitude as that of the incubation time proposed by Morozov and Petrov (2000), which shows that the incubation time plays a dominant role in the total initial delay time, and it also provides an reasonable explanation to the fundamental question pointed out by Clifton (Appl Mech Rev 46: 540-546, 1993).     

Aging, rejuvenation, and thixotropy in yielding magnetorheological fluids

The yielding behavior of dilute magnetorheological (MR) fluids has been investigated using creep–recovery tests. At very low stress levels, MR fluids behave in the linear viscoelastic regime as demonstrated by the fact that the instantaneous strain equals the instantaneous (elastic) recovery. In this region, gap-spanning field-induced structures support the stress levels applied. Upon increasing the stress value, the MR fluid evolves towards a nonlinear viscoelastic response. Here, the retarded elastic and viscous strain decrease, and the plastic contribution to the instantaneous strain grows probably due to the appearance of unattached field-induced structures. A larger stress value results in a viscoplastic solid behavior with negligible retarded and viscous strain and a fully plastic instantaneous strain. Finally, a plastic fluid behavior is found when the stress value is larger than the so-called yield stress. MR fluids exhibit an intermediate behavior between non-thixotropic (simple) and highly thixotropic model yield stress fluids.     

Motion and shape of an axisymmetric viscoplastic drop slowly falling through a viscous fluid

Slow sedimentation of a deformable drop of Bingham fluid in an unbounded Newtonian medium is studied using a variation of the integral equation method (Toose et al., J Eng Math 30:131–150, 1996, Int J Numer Methods Fluids 30:653–674, 1999). The Green function for the Stokes equation is used, and the non-Newtonian stress is treated as a source term. The computations are performed for a range of physical parameters of the system. It is demonstrated that initially deformed drop similar to Newtonian ones breaks up for high capillary number, Ca, and stabilizes to steady shapes at low Ca. Estimations of critical capillary number for specific initial deformations demonstrated its growth (increase in the stability of the drop) with the yield stress magnitude both for prolate and oblate initial shapes. Prolate initial shapes become more stable with the increase of the plastic viscosity. In contrast to this, for low yield stress, oblate shapes are destabilized with the growth of the plastic viscosity. This effect is similar to the effect of the viscosity of a Newtonian drop on its stability. However, at higher yield stress, the effect of plastic viscosity is reversed.