An important application of the variational formulation is to optimize the shear band thickness in strain localization processes. As an important application of the variational principles, we focus our attention in Chapter 3 on bumping localization problems.
Introduction
As an example, we then focus on the temperature update determined by the heat equation obtained from the first variation of the energy potential. To compare with the experimental observation, we present a simple one-dimensional model, and the temperature increases thus obtained agree well with experimental data [20].
General framework
P is the first Piola-Kirchhoff stress tensor; E is the internal energy per unit undistorted volume; In this case Z ={εp}, where εp is the effective plastic stress, and the Prandtl-Reuss flow rule for plastic deformation is required.
Variational form of the thermomechanical equations
The extension of the thermoelastic potential to general thermoelastic-viscoplastic materials follows the same line for thermoelastic materials. In particular, we are looking for a potential function for materials with uncoupled velocity sensitivity, viscosity, and thermal conductivity.
Variational thermomechanical updates
Equation (2.73c) gives the updated temperature as a function of the other state variables and corresponds exactly to (2.67c), written for timetn+1. Note that the factor T /Tn → 1 as 4t → 0, showing the consistency of the incremental equation (2.73d) with respect to its continuous equivalent.
Heat equation
Let us further assume that ²p is the only internal variable that coincides with the plastic strain in this case. The internal energy density is taken in a simple form. where η is the entropy, C is the elastic modulus, cv is the specific heat at constant volume, α is the coefficient of thermal expansion, θ is the absolute temperature, θ0 is the reference temperature, and the internal energy density is assumed to depend only on ²p through elastic deformation , i.e.
![Table 2.1: Material constants for α-titanium [40] and Al2024-T3.](https://thumb-ap.123doks.com/thumbv2/123dok/10401181.0/37.918.273.616.293.599/table-material-constants-α-titanium-40-al2024-t3.webp)
Energy-momentum tensor and configurational forces
As an example, we will consider the effect of temperature gradient on the propagation of a crack in a two-dimensional strip. As shown in (2.116), the existence of a temperature gradient introduces an additional thermal term in the traditional J integral. Since an increase of the J integral makes it closer to the fracture criterion JIC, we conclude that a positive temperature gradient accelerates the initiation and propagation of a crack, while a negative temperature gradient reduces the propagation of a crack.
Contrary to general arguments that the main effect of the temperature gradient on crack initiation is to make materials inhomogeneous with respect to their temperature-dependent mechanical properties, our result, however, is validated for any material including thermoelastic materials that have weak from temperature.
Summary and conclusions
It is followed by the introduction of the J integral and its application to crack propagation considering a strip with a certain temperature gradient.
Introduction
In this chapter, we consider strain localization strictly as a sub-grid phenomenon, and consequently the bands of strain localization are modeled as shear discontinuities. These shear discontinuities are confined to volume-element interfaces and are activated by the insertion of specialized strain localization elements. Kinematics of strain-locating elements is identical to kinematics of cohesive elements proposed by Ortiz and Pandolfi [21] for simulating fracture.
Unlike cohesive elements, the behavior of strain localization elements is governed directly by the same constitutive relation that governs the deformation of the volume elements.
General framework
The predictive ability of the approach is demonstrated using simulations by Guduruet al. One of the main objectives of this paper concerns the application of variational techniques to determine a. The determination of the effective incremental shear band energy φn(Fn+1k ,[[ϕn+1]]) involves optimization of the local internal state of the material within the band (cf. [12] and chapter 2) and optimization of the shear band thickness as well .
This statement supplies the configurational equilibrium and the additional field equation overS, which, provided that the stabilizing effect of heat conduction is properly taken into account, determines the local optimum value of the band thickness.
Adiabatic shear banding
As we shall see, the latter optimization can be regarded as a configurational balance statement consisting of the cancellation of the band-normal component of the Eshelby energy-momentum tensor. In the variational setting (cf. [12] and Chapter 2), this assumption simply means that Fp obeys the flow rule of the form. The reduction of the previous constitutive equations to the effective incremental form (3.8) requires the use of an update of the variational equation of state.
The shear band profile follows as a minimizer of the functional fn(ϕn+1, Tn+1, ²Pn+1,M) with appropriate boundary conditions at infinity.
Finite-element implementation
The natural coordinates (s1, s2) thus define a convenient curvilinear coordinate system for the middle surface of the element. Completing the formulation requires the calculation of the strain gradient and its derivative with respect to the node positions. For this purpose, let gα denote the tangent basis vectors in the deformed configuration and n the unit normal defined by gα.
It should be noted that the integral extends over the undeformed surface of the element in its reference configuration.
Simple-shear test
The evolution of the deformation field and the effective plastic deformation for a nominal deformation rate of 104 s-1 is shown in the figure. It is clear that the strain localization element captures the onset of the displacement discontinuity well. Since the mesh is relatively coarse and no mesh refinement is used to resolve the large deformation in the band, this actually shows a key advantage of using stress localization elements.
Moreover, the variational formulation of shear band thickness update proposed in the present model yields the optimized shear band thickness which is actually a function of deformation, as shown in Fig.
Dynamic shear bands
The minimum mesh size along the expected path of the shear band is 1 mm, and thus the thickness of the shear band is not selected by the mesh. The white line along the center of the shear stress concentration (blue) is the trace of the adiabatic shear band. In contrast, the calculated velocity of the shear band after initiation is in good agreement with experiment.
This highlights that the shear band thickness is variably updated throughout the simulation.
Summary and conclusion
It is interesting to note that the thickness grows steadily by an order of magnitude for the duration of the calculation. The thickness of the tape is determined as part of the solution using a variational principle. The deformation localization elements are equipped with full finite kinematics and are compatible with a conventional finite element discretization of the bulk material.
The versatility and predictive ability of the method is demonstrated through the simulation of dynamic shear band propagation.
Introduction
Based on the free volume model by Cohen and Grest (CG) [101], the viscosity can alternatively be expressed in terms of the average free volume per atom using a linear relationship between the free volume and the temperature [102]. Originated by Spaepen [106], the first hypothesis argues that this decrease is due to the formation of free volume and the introduced inhomogeneous flow is therefore controlled by the competition between. This hypothesis was then demonstrated by Argon [107], who analyzed the flow localization in a band of material by perturbing the strain rate due to the creation of free volume.
Following the finite deformation framework advanced by Ortiz and Stainier [12], we propose a constitutive model for bulk metallic glasses based on a free volume model.
General formulation
Following the theory of J2 plasticity, we assume that the flow rule governing the development of Fp is of the form. The volumetric elastic energy defined in (4.9) gives rise to a contribution of the free volume to the shape's deformation gradient. This term is an extension of the small strain definition introduced by Huang et al.
We assume that the deviatoric elastic strain energy density We,dev is quadratic in the deviatoric logarithmic elastic strain and is assumed to be of the form
Flow equation and kinetic potential
We assume that (4.25) can be extended to non-Newtonian flow, i.e. (4.18), by replacing the Newtonian viscosity with a non-Newtonian viscosity, while still retaining a double kinetic potential of the form (4.25). The non-linear behavior of the inverse kinetic relation is thus made implicit by means of the non-Newtonian viscosity. 4.7, a non-Newtonian viscosity and the quadratic double kinetic potential are sufficient to describe the nonlinear behavior of metallic glasses.
The existence of such a threshold voltage has previously been suggested by Taub, who attributed the threshold voltage to an increase in the internal energy of the material, similar to the way it is used for the free energy (4.2) in the present model.
Non-Newtonian viscosity
Free volume update
If free-volume diffusion is modeled by a free-space-like mechanism, then the net free-volume concentration rate is governed by a diffusion-production equation of the form [111]. The temperature dependence of the free volume concentration in the state of dynamic equilibrium above the glass transition temperature Tg (625 K for Vitreloy 1) can be approximately described by the linear Vogel-Fulcher relation. 120] proposed a threshold value, i.e. the free volume concentration at a temperature lower than the glass transition temperature is given by the constant ξ3p|T=Tg.
In addition, it is noteworthy that this expression also includes the dependence of the strain rate on the free volume.
Update algorithm
Finally, the total free volume concentration at state {σ, T} can be written as by considering both the effect of temperature and the net production of free volume. However, in the current model of metallic glasses, the temperature and the free volume are directly related by the universal expression for the free volume (4.39). It follows from (4.42) that the temperature increase in metallic glasses is updated through the development of the free volume and cannot be modeled separately, i.e. The Taylor-Quinney relation is not applicable to metallic glasses.
The resulting free volume concentrations (and temperatures) are then incorporated into the general formulation of the finite strain model described in Sect.
Experimental validation of Vitreloy 1 under uniaxial compres- sionsion
27, 98] at different strain rates and temperatures were used for the determination of the second type of parameters. The aim of the first validation is to reproduce the quasi-static constitutive behavior of Vitreloy 1 at room temperature reported previously by Bruck et al. However, the steady state of the flow stress at room temperature was not observed in the experiments due to the formation of shear bands, which caused the failure of the specimens.
Thus, we conclude that an increase in free volume leads to strain relaxation in bulk metallic glasses, while an increase in temperature (corresponding to an increase in free volume) accelerates strain localization and material failure.
Shear banding of metallic glass plate under bending
The excellent agreement between the present model and experimental data (Fig. 4.4–Fig. 4.10) shows that our continuum model very well captures the main features of the behavior of bulk metallic glasses. Moreover, it is noteworthy that the model can be easily implemented in a finite element code, which enables efficient numerical simulation of the dynamic behavior of bulk metallic glasses, as shown in the next section. This, together with the development of variational strain localization finite elements (cf. chapter 3), makes it possible to perform finite element simulation of the shear band of bulk metallic glasses.
The shear band induced displacement jumps are easily visible, resulting in an average shear band spacing of 90µm, which is 15% of the plate thickness.
Summary and conclusions
An experimental study of the formation process of adiabatic shear bands in a structural steel. Journal of the Mechanics and Physics of Solids, 36:251–end, 1988. Analysis of the dynamic propagation of adiabatic shear bands. International Journal of Solids and Structures. Deformation behavior of bulk Zr41.2Ti13.8Cu12.5Ni10Be22.5 metallic glass over a wide range of strain rates and temperatures.
Thermodynamics and kinetics of bulk metallic glass-forming liquid Zr41.2Ti13.8Cu12.5Ni10Be22.5: glass formation from a solid liquid.
![Figure 2.3: Stress-strain curves of Al2024-T3 at strain rate 1 s −1 and 3000 s −1 . Experimental data are taken from [37].](https://thumb-ap.123doks.com/thumbv2/123dok/10401181.0/39.918.274.612.188.496/figure-stress-strain-curves-al2024-strain-experimental-taken.webp)
