Derivation of equations of state for ideal crystals of simple structure

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtaining the equation of state. Earlier, this approach was used to obtain the equation of state in the Mie-Grüneisen form for a one-dimensional lattice. The aim of this paper is to develop and generalize this approach to the three-dimensional case. As a result, we obtain the dependence of the Grüneisen function on the volume, which is compared with the computations performed according to well-known models with experimental data taken into account. It is proved that the Grüneisen coefficient substantially depends on the form of the strain state. Moreover, we refine the equation of state; namely, we show that the Grüneisen coefficient depends on the thermal energy, but this dependence in the three-dimensional case is much weaker than in the one-dimensional case. A refined equation of state containing a nonlinear dependence on the thermal energy is obtained     

Thermodynamic relations along the principal Hugoniot

Some thermodynamic relations are derived along the principal Hugoniot of materials for which the Grüneisen relation is a function of volume only. Rather than being expressed in terms of traditional thermodynamic variables, such as volume and temperature, the relations are expressed in terms of the shock-Hugoniot behavior and of a term grouping that is related to the Grüneisen function. By so doing, a new perspective is gained on the interrelation of thermodynamic quantities along the Hugoniot, which may be of use in developing analytical equations of state. Received 3 November 1997 / Accepted 30 April 1998     

An equation of state for dense nitrogen

An empirical equation of state for nitrogen at high pressure and density is considered. It is shown that for nitrogen at densities greater than 0.6 g/cm³, by using available data [1–3] on static compression of gaseous nitrogen and shock compression of liquid nitrogen, it is possible to construct a Mie-Grüneisen type equation of state which gives a pressuredensity relationship close to experiment along the shock adiabat of liquid nitrogen and agrees with the calculations of other authors for temperature values beyond the shock-wave front [2–3]. Heat capacity, entropy, and Grüneisen coefficient values beyond the shock-wave front in liquid nitrogen are calculated.     

Penetration of Cylindrical Projectiles into Saturated Sandy Media

The time and depth of vertical one-dimensional projectile penetration into sandy media in the near shore region are derived. A precise definition for the physical properties and for the behavior of the sandy medium following the projectile impact are evaluated. Three separate time intervals following projectile impact are identified. During the first 3 ms of penetration, the deviatoric friction stress is shown to be negligible and the integrated Mie–Grüneisen equation of state (or, equivalently, the Hugoniot-adiabat) may be applied to compute the normal penetration resistance force from the sand pressure. In order to compute sand pressure as a function of the sand density D by the integrated Mie–Grüneisen equation of state, the Mie–Grüneisen dimensionless constants γ₀ and s and the dimensional speed of sound C ₀ in the sandy medium are required. In order to illustrate the one-dimensional shock wave propagation in both wet and dry sands, Hugoniot data for wet and dry silica sands are evaluated by a three degrees of freedom algorithm to compute these required constants. The numerical results demonstrate that the amplitude of the shock wave pressure in the wet silica sand (41% porosity) is approximately one-third of the shock wave pressure amplitudes in the dry silica sands (22% and 41% porosity). In addition, the shock wave pressure dampens quicker in the wet sand than in the dry sands.     

疏松金属材料冲击温度理论分析

以热力学原理和固态物质的三项式物态方程为基础,由密实物质的冲击绝热线和热力学状态,通 过等容线法推导出了疏松金属材料的冲击温度理论计算方法。以铁为例,分析了几种物理参数对该模型计算 结果的影响。计算和分析结果显示,利用新模型得到的计算结果与已有实验结果吻合较好,误差均在5%以 内。疏松金属材料的冲击温度受Grneisen系数、电子Grneisen系数影响不大,而密实度、冲击压力和电子 比热系数则会对疏松金属材料冲击温度产生较大影响。     

Thermomechanical constitutive equations for the dynamic response of ceramics

The behavior and failure of brittle materials is significantly influenced by the existence of inhomogeneities such as pores and cracks. The proposed constitutive equations model the coupled micro-mechanical response of these inhomogeneities through evolution equations for scalar measures of porosity, and a “density” function of randomly oriented penny-shaped cracks. A specific form for the Helmholtz free energy is proposed which incorporates the known Mie–Grüneisen constitutive equation for the nonporous solid. The resulting thermomechanical constitutive equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. These equations allow for the simulation of the following physical phenomena exhibited by brittle materials: (1) high compressive strength compared with much lower tensile strength; (2) inelastic deformation due to growth and nucleation of cracks and pores instead of due to dislocation dynamics associated with metal plasticity; and (3) loss of integrity (degradation of elastic moduli) due to damage accumulation. The main features of the model are demonstrated by examples of cyclic loading in homogeneous deformation and by a simulation of a dynamic plate-impact experiment on AD85 ceramic. The theoretical predictions of the model are in excellent agreement with the dynamic experimental data.     

The influence of microstructural arrangement on the response of polymer composites in planar impact

In recent years there has been an increased demand for advanced materials that can sustain rapid dynamic loadings. To this end, we simulate the transient response of composites with nonuniform arrangements of their microstructures. First, a constitutive model that reproduces experimentally measured response of a glass-fiber composites is identified and adjusted. This involves a Mie–Grüneisen equation of state for the dilatational response together with a Voigt model for the isochoric behavior which is modified to include damage effects from void nucleation and growth. Then, with the aid of this constitutive model, a sequence of simulations of composites with nonuniform distributions of the reinforcement are executed. We find that composites with increasing volume fraction of the reinforcement along the impact direction tend to attenuate the intensity of the propagating waves. This attenuation delays the initiation of failure mechanisms to higher impact velocities and improves the composite’s sturdiness.     

Calculation of Hugoniot properties for shocked nitromethane based on the improved Tsien’s EOS

We have calculated the Hugoniot properties of shocked nitromethane based on the improved Tsien＇s equa- tion of state （EOS） that optimized by ＂exact＂ numerical molecular dynamic data at high temperatures and pressures. Comparison of the calculated results of the improved Tsien＇s EOS with the existed experimental data and the direct simu- lations show that the behavior of the improved Tsien＇s EOS is very good in many aspects. Because of its simple analytical form, the improved Tsien＇s EOS can be prospectively used to study the condensed explosive detonation coupling with chemical reaction.     

On shock waves in a special class of thermoelastic solids

This paper describes a thermoelastic model for shock waves in uniaxial strain based on a subclass of the so-called materials of Mie–Grüneisen type. We compare the Hugoniot curve with the isotherms and isentropes for this model, and we construct the shock-wave solution to a simple impact problem.     

On the behavior of fine mud suspensions

Flows of natural mud-water mixtures are of great interest for industrial and civil engineering. But there is still no general agreement about the methods for determining the main rheological characteristics of these systems. We propose here an accurate rheological study of some natural mud-water mixtures. We first discuss the possible effects of changing various parameters such as temperature, pH, electrolyte concentration, solid concentration, clay type. The behavior of these muds appears to be very sensitive to most of these parameters and to be hardly predictable from a knowledge of their components. Then, we show that a Herschel-Bulkley model fits very well steady flow experimental data for a very large range of shear rates. We also suggest physical explanations of this model in agreement with our observations of behavior changes when some parameters change. The yield stress value of this model provides a good estimation of real yield stress which is a key parameter for mixture behavior. These considerations are very useful to characterize, predict, and compare various mud flows.     

Multi-scale defect interactions in high-rate brittle material failure. Part I: Model formulation and application to ALON

Within this two part series we develop a new material model for ceramic protection materials to provide an interface between microstructural parameters and bulk continuum behavior to provide guidance for materials design activities. Part I of this series focuses on the model formulation that captures the strength variability and strain rate sensitivity of brittle materials and presents a statistical approach to assigning the local flaw distribution within a specimen. The material model incorporates a Mie–Grüneisen equation of state, micromechanics based damage growth, granular flow and dilatation of the highly damaged material, and pore compaction for the porosity introduced by granular flow. To provide initial qualitative validation and illustrate the usefulness of the model, we use the model to investigate Edge on Impact experiments (Strassburger, 2004) on Aluminum Oxynitride (AlON), and discuss the interactions of multiple mechanisms during such an impact event. Part II of this series is focused on additional qualitative validation and using the model to suggest material design directions for boron carbide.     

Thermo-mechanical constitutive modeling of shape memory polyurethanes using a phenomenological approach

This study examined the constitutive modeling of shape memory polyurethanes (SMPUs). SMPUs exhibit a thermo-responsive shape memory behavior, i.e., a thermally fixed temporary shape at a low temperature that returns to its original (permanent) shape when heated. This unique property arises from the molecular configuration of their hard and soft segments; the latter can form a variable state ranging from a rubbery (active) to rigid (frozen) phase according to temperature, while the former undergoes little deformation and acts as a fixed net between the soft segments. In this study, a three-phase phenomenological model (one hard segment phase and two (active and frozen) soft segment phases) was developed to describe the deformation behavior of SMPUs according to their microstructure. The stress and strain relationships of each phase are described mathematically using one three-element viscoelastic and two Mooney–Rivlin hyperelastic equations, respectively. The total stress was calculated by combining those equations via some internal variables that can track the volume fractions of the active and frozen phases and a non-mechanical frozen strain. For validation, the cyclic thermo-mechanical behavior of a SMPU was predicted. These predictions were compared with the experimental results with reasonable agreement between them.     

A constitutive model in linear thermoviscoelasticity of polymers based on the concept of cooperative relaxation

New constitutive relations are derived for amorphous glassy polymers based on the concept of cooperative relaxation. A polymer is treated as a system of rearranging regions (flow units) embedded into a homogeneous elastic matrix. The viscoelastic (time-dependent) response of a medium is explained by rearrangements of segments of long chains in relaxing regions which occur at random instants. The kinetics of rearrangement is described in the framework of the Eyring concept of thermally activated processes, whereas the energy of any flow unit is assumed to randomly change at the instant of its reformation. Based on experimental data, phenomenological formulas are proposed for material functions. Adjustable parameters are found by fitting observations for mixtures of nylon with lithium halides in isothermal tensile relaxation tests. The thermoviscoelastic response in other tests is studied numerically. It is demonstrated that the material behavior predicted by the constitutive model in non-isothermal tests substantially differs from that predicted by conventional models whose adjustable parameters are determined by using the same experimental data. Received September 30, 1998     

Mode I Fracture Characterization of Bituminous Paving Mixtures at Intermediate Service Temperatures

This study presents an integrated approach combining experimental tests and numerical modeling to characterize mode I fracture behavior of bituminous paving mixtures subjected to a wide range of loading rates at intermediate temperature conditions. A simple experimental protocol is developed using the semi-circular bending (SCB) test geometry. The local fracture behavior at the initial notch tip of the SCB specimens is monitored using high-speed cameras with a digital image correlation (DIC) system. The DIC results of the SCB fracture tests are then simulated using a finite element method that is incorporated with material viscoelasticity and cohesive zone fracture. Fracture properties are obtained locally at the notch tip by identifying two cohesive zone fracture parameters (cohesive strength and fracture energy) that result in a good agreement between test results and numerical simulations. The results clearly present significant rate-dependent fracture characteristics of bituminous paving mixtures at intermediate service temperatures. This study further demonstrates that fracture properties of viscoelastic materials need to be characterized at the local fracture process zone when they present ductile fracture behavior.     

On the relation between particle velocity and shock wave speed for thermoelastic materials

Results of shock-wave experiments in solids often suggest a nearly-linear relation between the particle velocity behind the shock and the shock wave speed. The present note reconsiders the question of whether thermoelastic material models may be consistent with such observations. Emphasis is placed on the role played by the response of the material in severe compression, as distinguished from its response for small or moderate deformations. The details are illustrated for materials of Mie-Grüneisen type. Received 21 December 2001 / Accepted 22 April 2002 Published online 8 July 2002     

On the jet collision: General model and reduction to the Mie-Grüneisen state equation

We study the stationary direct supersonic collision of jets of condensed materials. We determine the basic flow characteristics: the maximum values of pressure, temperature, and densities on outgoing shock wave fronts and at the wave stagnation and penetration points. To this end, just as in the Lavrentiev problem about the jet collision in the framework of an incompressible fluid model, it suffices to consider the flow only along the central streamline, i.e., the symmetry axis. We consider the general caloric (incomplete) equation of state and, to close the thermodynamic construction and determine the temperature dependence on the state parameters, supplement them with thermodynamic identities. We also consider the conditions on discontinuities, the Bernoulli integrals, i.e., the conservation laws, to relate the states behind the wave front and the stagnation point, and the continuity conditions at this point. Just as in the collision problem for jets of incompressible fluid, we neglect the strength, viscosity, and heat conduction. As a result, we construct a mathematical model, i.e., a system of 12 integro-algebraic equations, and propose a semi-inverse solution method, in which the system splits into separate equations. In the special case of the Mie-Grüneisen state equation, the system becomes much simpler. We perform computations and construct the dependence of maximal pressures and temperatures on the impact velocity in the range 1–20 km/s for many pairs of materials of the colliding jets. We also compare the results with the solution obtained according to the incompressible fluid model.     

A simple semiempirical model for the effective viscosity of multicomponent suspensions

We propose a methodology to approximate the viscosity of multicomponent suspensions. The procedure consists of successive applications of expressions for the viscosity of binary mixtures, originally written as the product of monomodal stiffening functions. First, the viscosity of a binary mixture made of the two smallest components is calculated. This allows to extract a volume fraction that will be used, together with the volume fraction of the third component, to feed the next iteration of the procedure to calculate the viscosity of a trimodal mixture and so on. The application of this approach to arbitrary mixtures requires the detailed knowledge of the geometry of the system in the form of size ratios and compositions. When this information is unknown, an approximation of the model can still be used as a fitting tool. With that purpose, the final expression for the viscosity is written in terms of an effective volume fraction that is further approximated by the use of a (1,2) Padé approximant. This approximation allows to incorporate the crowding effects due to different species in a volume fraction-dependent crowding factor that can be used as a fitting parameter to match experimental or simulation data. We have applied the model to mixtures of particles with different sizes and tested its accuracy comparing with experimental results obtaining very good agreement.