Short-Term Microdamage of a Granular Material under Physically Nonlinear Deformation

The structural theory of short-term damage is generalized to the case where the undamaged components of a granular composite deform nonlinearly. The basis for this generalization is the stochastic elasticity equations for a granular composite with porous components whose skeletons deform nonlinearly. Microvolumes of the composite components meet the Huber–Mises failure criterion. Damaged microvolume balance equations are derived for the physically nonlinear materials of the components. Together with the equations relating macrostresses and macrostrains of a granular composite with porous nonlinear components, they constitute a closed-form system. The system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the microdamage–macrostrain relationship and plotting deformation diagrams are proposed. Uniaxial tension curves are plotted for the case where microdamages occur in the linearly hardened matrix and do not in the inclusions, which are linearly elastic     

Theory of Short-Term Microdamageability of Granular Composite Materials

The theory of microdamageability of granular composites is outlined through the simulation of microdamages in the components by pores filled with compression-resisting particles of a destroyed material. The damage criterion for a microvolume of a component is taken in the Schleicher–Nadai form, which allows for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of Weibull-distributed coordinates. The stress–strain state and the efficient properties of the material are determined from the stochastic equations of elastic theory for a granular composite with porous components. The equations of deformation and microdamage are closed by the equations of porosity balance in the components. Nonlinear diagrams of the concurrent processes of deformation in the granular material and microdamage in the matrix are plotted. The effect of the physical and geometrical parameters on them is studied     

Influence of Physically Nonlinear Deformation on Short-Term Microdamage of a Laminar Material

The structural theory of short-term damage is generalized to the case where the undamaged components of an N-component laminar composite deform nonlinearly. The basis for this generalization is the stochastic elasticity equations for an N-component laminar composite with porous components whose skeleton deforms nonlinearly. Microvolumes of the composite components meet the Huber–Mises failure criterion. Damaged microvolume balance equations are derived for the physically nonlinear materials of the composite components. Together with the equations relating macrostresses and macrostrains of the laminar composite with porous nonlinear components, they constitute a closed-form system. This system describes the coupled processes of physically nonlinear deformation and microdamage. For a two-component laminar composite, algorithms for calculating the microdamage–macrostrain relationship and plotting deformation curves are proposed. Uniaxial tension curves are plotted for the case where microdamages occur in the linearly hardening component and do not in the linearly elastic component     

Short-term microdamage of a physically nonlinear particulate material under a combination of normal and tangential loads

The structural theory of short-term damage is generalized to the case where the undamaged components of a particulate composite deform nonlinearly under loads that induce a compound stress state. The basis for this generalization is the stochastic elasticity equations for a particulate composite with porous components whose skeletons deform nonlinearly. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. Balance equations for damaged microvolume are derived for the physically nonlinear materials of the components. Together with the macrostress-macrostrain relationship for a particulate composite with porous nonlinear components, they constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the microdamage-macrostrain relationship and plotting stress-strain curves are proposed. Such curves are plotted for the case where the composite is subjected to a combination of normal and tangential loads, and microdamages occur in the linearly hardened matrix and do not in the linearly elastic inclusions. The stress-strain curves are examined depending on the volume fraction of inclusions and presence of tangential stresses __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 48–57, December, 2006.     

Deformation of a laminated composite with a physically nonlinear reinforcement and microdamageable matrix

The structural theory of short-term microdamage is generalized to a laminated composite with a microdamageable matrix and physically nonlinear reinforcement. The basis for the generalization is the stochastic elasticity equations of a laminated composite with a porous matrix. Microvolumes in the matrix material meet the Huber-Mises failure criterion. The damaged-microvolume balance equation for the matrix is derived. This equation and the equations relating macrostresses and macrostrains of a laminated composite with porous matrix and physically nonlinear reinforcement constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage occurring in different composite components. Algorithms for computing the microdamage-macrostrain relationships and deformation diagrams are developed. Uniaxial tension curves are plotted for a laminated composite with linearly hardening reinforcement __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 11, pp. 47–56, November 2005.     

Deformation of a fibrous composite with physically nonlinear fibers and microdamageable matrix

The structural theory of short-term microdamage is generalized to a fibrous composite with a microdamageable matrix and physically nonlinear fibers. The basis for the generalization is the stochastic elasticity equations of a fibrous composite with a porous matrix. Microvolumes in the matrix material meet the Huber-Mises failure criterion. The damaged-microvolume balance equation for the matrix is derived. This equation and the equations relating macrostresses and macrostrains of a fibrous composite with porous matrix and physically nonlinear fibers constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage occurring in different components of the composite. Algorithms for computing the microdamage-macrostrain and macrostress-macrostrain relationships are developed. Uniaxial tension curves are plotted for a fibrous composite with linearly hardening fibers __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 38–47, January 2006.     

A Note on the Theory of Short-Term Microdamageability of Granular Composites under Thermal Actions

The theory of microdamageability of granular composites is stated with allowance made for the thermal effect. Microdamages in the components are modeled by pores, hollow or filled with particles of the destroyed material that resist compression. The fracture criterion is assumed to have the Schleicher–Nadai form, which takes into account the difference between the tensile and compressive ultimate loads, with the ultimate strength being a random function with a power or Weibull distribution. The stress–strain state and effective properties of the material are determined from the stochastic thermoelastic equations for granular composites with porous components. The equations of deformation and microdamage are closed by the equation of porosity balance corrected for the thermal effect. Nonlinear diagrams are plotted for the concurrent processes of deformation of a granular material and microdamage of the matrix as functions of macrostrains and temperature. The influence of the physical and geometrical parameters on the processes is analyzed.     

Short-term microdamage of laminated material with nonlinear matrix and microdamaged reinforcement

A structural theory of short-term microdamage is proposed for a two-component laminated composite with microdamageable reinforcement and physically nonlinear matrix. The basis of the theory is the stochastic elasticity equations of a laminated composite with a porous reinforcement. Microvolumes in the reinforcement material meet the Huber-Mises failure criterion. The damaged-microvolume balance equation for the reinforcement is derived. This equation and the equations relating macrostresses and macrostrains of a laminated composite with porous reinforcement and physically nonlinear matrix constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage occurring in different composite components. Algorithms for computing the microdamage-macrostrain relationships and deformation diagrams are developed. Uniaxial tension curves are plotted for a laminated composite with linearly hardening matrix __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 3–12, December 2005.     

Flow through isotropic granular porous media

A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation.     

Upscaling Forchheimer law

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.     

Short-term microdamageability of a fibrous composite with physically nonlinear matrix and microdamaged reinforcement

A structural theory of short-term microdamage is proposed for a fibrous composite with physically nonlinear matrix and microdamaged reinforcement. The theory is based on the stochastic elasticity equations of a fibrous composite with porous fibers. Microvolumes of the fiber material are damaged in accordance with the Huber-Mises failure criterion. A balance equation for damaged microvolumes in the reinforcement is derived. This equation together with the equations relating macrostresses and macrostrains of a fibrous composite with porous reinforcement and physically nonlinear matrix constitute a closed-form system. This system describes the coupled processes of physically nonlinear deformation and microdamage that occur in different components of the composite. Algorithms are proposed for computing the dependences of microdamage on macrostrains and macrostresses on macrostrains. Uniaxial tension curves are plotted for a fibrous composite with a linearly hardening matrix __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 3–13, February 2006.     

非平整、多孔介质海底上波浪传播的复合方程

为了反映近岸区域实际存在的多孔介质海底效应，并且考虑到波浪在刚性海底上传播模型的 最新研究进展，运用Green第二恒等式建立了波浪在非平整、多孔介质海底上传播的复合方 程. 假设水深和多孔介质海底层厚度均由两种分量组成：慢变分量，其水平变化的长度尺度大于 表面波的波长；快变分量，其水平变化的长度尺度与表面波的波长等阶，但其振幅小于表面 波的振幅. 另外，多孔介质层下部边界的快变分量比水深的快变分量小1个量级. 针对水体层和多孔介质层，选择Green第二恒等式方法给出了波浪传播和渗透的复合方程， 它在交接面上满足压力和垂直渗透速度的连续性条件,可充分考虑波数变化的一般连续性， 并包含了某些著名的扩展型缓坡方程.     

Nonlinear deformation of a particulate material in complex stress state

An algorithm is proposed to determine the effective deformation properties and stress-strain state of particulate composite materials with physically nonlinear components and complex stress state. The laws that govern the deformation of particulate composites are studied. A particulate composite is considered a two-component material of random structure. Its effective properties are determined by conditional averaging. The nonlinear equations that incorporate the physical nonlinearity of the components are solved by the method of successive approximations. The relationship between macrostresses and macrostrains is established. The effective deformation properties of a particulate composite as a function of the volume fractions of the components and stress state are studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 50–60, March 2006.     

Elasto-viscoplastic-viscous theories of granular and porous media

The mechanics of granular and porous media is considered in the light of the modern theories of structured continuum. The basic laws of motion are presented and several constitutive relations are derived. The special case of elastic porous media is considered in detail and the basic field equations are derived and the possible application of the results to soil dynamics is pointed out. The theory of the flow of granular media is also considered and basic equations of motion are derived where the results of Goodman and Cowin are recovered. The viscoplastic flow of porous media is studied and the possible application to soil mechanics is also discussed.     

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.     

First-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations

A procedure for studying the first-passage failure of strongly non-linear oscillators with time-delayed feedback control under combined harmonic and wide-band noise excitations is proposed. First, the time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. A backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is worked out in detail to illustrate the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed. The validity of the proposed method is confirmed by digital simulation.     

The Effective Elastic Constants of Porous Polycrystals of Trigonal Symmetry

A theory behind the effective properties of piezoelectric crystals is considered. It is based on the stochastic electroelastic equations of a microinhomogeneous medium. The initial equations are reduced to integral equations, which, after conditional averaging, are reduced to a system of linear algebraic equations in component-average parameters. The effective electroelastic properties of porous polycrystals with trigonal symmetry are evaluated     

Deformation of physically nonlinear stochastic composites

The studies of the deformation of physically nonlinear homogeneous and composite materials are systematized. Algorithms to determine the effective elastic properties and stress–strain state of particulate, laminated, fibrous, and laminated fibrous composite materials with physically nonlinear components are outlined, and their deformation patterns are studied. Composites are considered as two-component materials of random structure. Their effective properties are determined using the conditional averaging method. The nonlinear equations that allow for the physical nonlinearity of the components are solved by an iterative method. The relationship between macrostresses and macrostrains is established. Macrostress–macrostrain curves of homogeneous and composite materials are analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 7–38, December 2008.     

Simulation of the Short-Term Microdamageability of Laminated Composites

The theory of microdamageability of multicomponent laminated composites is outlined through the simulation of microdamages in the components by pores filled with compression-resisting particles of the destroyed material. The damage criterion for a microvolume of a component is taken in the Schleicher–Nadai form, which allows for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of Weibull-distributed coordinates. The stress–strain state and the efficient properties of the material are determined from the stochastic equations of the elastic theory for a laminated composite with porous components. The equations of deformation and microdamage are closed by the equations of porosity balance in the components. Nonlinear diagrams of the concurrent processes of deformation in the laminated material and microdamage in the matrix are plotted. The effect of the physical and geometrical parameters on them is studied     

Short-Term Microdamageability of Fibrous Composites with Transversally Isotropic Fibers and a Microdamaged Binder

The theory of microdamageability of fibrous composites with transversally isotropic fibers and a microdamaged isotropic porous matrix is proposed. Microdamages in the matrix are simulated by pores filled with particles of the destroyed material that resist compression. The criterion of damage in the matrix microvolume is taken in the Schleicher–Nadai form. It accounts for the difference between the ultimate tensile and compressive loads. The ultimate strength is a random function of coordinates with Weibull distribution. The stress–strain state and effective properties of the material are determined from the stochastic equations of the elastic theory for a fibrous composite with porous components. The equations of deformation and microdamage are closed by the equations of porosity balance in the matrix. Nonlinear diagrams of the concurrent processes of deformation of fibrous materials and microdamage of the matrix are plotted. The effect of the physical and geometrical parameters on them is studied