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.     

Elastic properties and long-term damage of transversely isotropic composites with stress-rupture microstrength described by a fractional-power function

The damage process is modeled by randomly dispersed micropores occurring in places of destroyed microvolumes according to the stress-rupture microstrength, which is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, according to the Huber–Mises criterion, and is a random function of coordinates. Given microstresses or microstrains, the equations of porosity balance at an arbitrary time are derived. Together with the macrostress–macrostrain relationships for a discrete fibrous composite with porous components, they describe the coupled processes of deformation and long-term damage. A specific problem with a bounded stress-rupture microstrength function is solved Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 71–81, January 2009.     

Theory of Short-Term Microdamageability for a Homogeneous Material under Physically Nonlinear Deformation

The structural theory of short-term damageability is generalized to the case of physically nonlinear deformation of an undamaged material. The stochastic elasticity equations for a porous medium whose skeleton deforms nonlinearly are used. The failure criterion for a microvolume of the material is assumed to be in the Huber–Mises form. The microdamage balance equation for a physically nonlinear material is derived. This equation and the macrostress–macrostrain relation for a porous physically nonlinear material constitute a closed-form system describing the coupled processes of physically nonlinear deformation and microdamage. An algorithm is constructed for computing microdamage–macrostrain relationships and plotting deformation curves. Such curves are plotted for the case of uniaxial tension     

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     

Micromechanics of long-term damage of particulate composites with unlimited microdurability

The theory of long-term damage of homogeneous materials is generalized to particulate composite materials. The damage of the composite components is modeled by randomly dispersed micropores. The damage criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the tensile strength, according to the Huber–Mises criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the composite components at an arbitrary time is formulated. Algorithms of calculating the time dependence of microdamage and macrostresses or macrostrains are developed and relevant curves are plotted in the case of unlimited microdurability Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 7–17, November 2008.     

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     

Homogenization and Global Responses of Inhomogeneous Spherical Nonlinear Elastic Shells

Homogenization of radially inhomogeneous spherical nonlinear elastic shells subject to internal pressure is studied. The equivalent homogeneous material is defined in such a way that it gives rise to exactly the same global response to the pressure load as that of the inhomogeneous shell. For a shell with general strain–energy function and inhomogeniety, the strain–energy function of the equivalent homogeneous material is determined explicitly. The resulting formula is used to study layered composite shells. The equivalent homogeneous material for an infinitely fine layered composite shell is examined, and is found to give not only the same global response, but also the same average stress field as the composite shell does.     

Deformation and long-term damage of orthotropic composites with limited stress-rupture microstrength

The theory of long-term microdamage of homogeneous materials based on the mechanics of stochastically inhomogeneous materials is generalized to a composite with orthotropic inclusions. The damage of the composite components is modeled by randomly dispersed micropores. The damage criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the tensile strength, according to the Huber–Mises criterion, and assumed to be a random function of coordinates. Given macrostresses or macrostrains, an equation of damage (porosity) balance in the composite components at an arbitrary time is derived. The time dependence of microdamage and macrostresses or macrostrains in a discrete-fiber-reinforced composite with limited stress-rupture microstrength described by a fractional-power function is plotted     

Stability of plates and shells made of homogeneous and composite materials subject to short-term microdamage

The paper analyzes results on the bifurcation buckling of plates and shells of revolution made of homogeneous and composite materials subject to progressive damage __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 3–35, March 2008.     

Deformation and long-term damage of particulate composites with stress-rupture microstrength described by a fractional-power function

The theory of long-term damage of homogeneous materials is generalized to particulate composite materials. The damage of the composite components is modeled by randomly dispersed micropores. The damage criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the tensile strength, according to the Huber-Mises criterion, and assumed to be a random function of coordinates. An equation of damage (porosity) balance in the composite components at an arbitrary time is formulated. Algorithms of calculating the time dependence of microdamage and macrostresses or macrostrains are developed and corresponding curves are plotted in the case of limited microdurability Translated from Prikladnaya Mekhanika, Vol. 44, No. 10, pp. 3–12, October 2008.     

Coupled processes of deformation and damage of composites with orthotropic inclusions and unbounded rupture-stress function

The theory of long-term damage of homogeneous materials, which is based on the equations of the mechanics of stochastically inhomogeneous materials, is generalized to composite materials reinforced with orthotropic ellipsoidal inclusions. The microdamage of the composite components is modeled by randomly dispersed micropores. The failure criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit, which is the tensile strength, according to the Huber–Mises criterion, and assumed to be a random function of coordinates. Given macrostresses or macrostrains, an equation of porosity balance in the composite components at an arbitrary time is formulated. The time dependence of microdamage and macrostresses or macrostrains is established in the case of unlimited stress-rupture microstrength described by an exponential power function     

Long-term damage of discrete-fiber-reinforced composites with transversely isotropic inclusions and stress-rupture microstrength described by an exponential power function

The theory of long-term damage of homogeneous materials, which is based on the equations of the mechanics of stochastically inhomogeneous materials, is generalized to discrete-fiber-reinforced composite materials. The microdamage of the composite components is modeled by randomly dispersed micropores. The failure criterion for a microvolume is characterized by its stress-rupture strength. It is determined by the dependence of the time to brittle failure on the difference between the equivalent stress and its limit. Given macrostresses and macrostrains, an equation of damage (porosity) balance in the composite components at an arbitrary time is formulated. The time dependence of microdamage and macrostresses or macrostrains is established in the case of stress-rupture microstrength described by an exponential power function Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 19–29, February 2009.     

General hoyle–youngdahl and love solutions in the linear inhomogeneous theory of elasticity

The general Hoyle–Youngdahl and Love solutions in the three-dimensional theory of inhomogeneous linear elastic materials are proposed. Following a brief historical outline of various general solutions existing in the classical linear elasticity of homogeneous isotropic media, key steps of the derivation of the Hoyle–Youngdahl and Love solutions are presented. The procedure is then generalized to the case of inhomogeneous elastic materials with elastic constants depending on the z-coordinate. The significance of the solutions and their relevance to modeling of functionally graded materials is discussed in brief     

Strain Energy Functions for a Poisson Power Law Function in Simple Tension of Compressible Hyperelastic Materials

The most general strain energy function that yields a power law relationship between the principal stretches in the simple tension of nonlinear, elastic, homogeneous, compressible, isotropic materials is obtained. The approach taken generalises that used by Blatz and Ko. The strain energy function obtained depends on the choice of two stretch invariants. The forms of the strain energy function for a number of such choices are obtained. Finally, some consequences of the choice of strain energy function on the stress–strain relationship for uniaxial tension are investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamic deformation of glued beams

The dynamic deformation of two glued rectangular beams is studied. A problem formulation and problem-solving method are given. It is assumed that the structure moves in a vertical plane. The materials of the beams and adhesive interlayer (AI) are homogeneous and isotropic. The geometrical and mechanical characteristics of the beams are different. Assumptions on the stress–strain state of the beams and AI are formulated based on the theory of elasticity. The composite beam is decomposed into three beams. The virtual-displacement principle is formulated for each of them. The problem is solved by finding five functions defined on the beam axis. The formulated principles allow solving different problems of dynamic deformation of glued beams     

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.     

Mircopolar Model of Fracture for Composite Material

Failure of a composite is a complex process accompanied by irreversible changes in the microstructure of the material. Microscopic mechanisms are known of the accumulation of damage and failure of the type of localized and multiple ruptures of the fibers delamination along interphase boundaries, and also mechanisms associated with fracture of fibers. In this work, we propose a mathematical model of the local mechanism of failure of a composite material randomly reinforced with a system of short fibers. We implement the Cosserat moment model of crack tip for filament material, reinforced with whiskers or in fiber- reinforced polycrystalline materials. It is assumed that the angular distribution of the fibers is isotropic and the elastic characteristics of the fibers are considerably higher than the elastic constants of the matrix. We implement the homogenization procedure for the effective Cosserat constants similarly to the effective elastic constants. The singular solution in the vicinity of the crack tip in the Cosserat moment model is found. Using this solution, we examine the bending stresses in the filaments due to effective moment stresses in the material. The constructed model describes the phenomenon of fracture of the fibers occurring during crack propagation in those composites. The following assumptions are used as the main hypotheses for the micromechanical model. The matrix contains a nucleation crack. When the load is increased the crack grows and its boundary comes into contact with the reinforcing fibers. A further increase of the stress causes bending of the fiber. When~the fiber curvature reaches a specific critical value, the fiber ruptures. If the stress at infinity is given, the fibers no longer delay the development of failure during crack propagation The degree of bending distortion of the fiber in the vicinity of the boundary of the crack is determined by the moment model of the material. The necessity to take into account the moment stresses in the failure theory of the reinforced material was stressed in [Muki and Sternberg (1965) Zeitschrift f angew Math und Phys 16:611–615; Garajeu and Soos (2003) Math Mech Solids 8(2):189–218; Ostoja-Starzewski et al (1999) Mech Res Commun 26:387–396]. The moment Cosserat stresses were accounted also for inhomogeneous biomechanical materials by Buechner and Lakes (2003) Bio Mech Model Mechanobiol 1: 295–301. We should also mention the important methodological studies [Sternberg and Muki (1967) J Solids Struct 1:69–95; Atkinson and Leppington (1977) Int J Solids Struct 13: 1103–1122] concerned with the moment stresses in homogeneous fracture mechanics.     

Averaging of an orthotropic elastic plate weakened by periodic hinges of finite stiffness

A formal asymptote of a solution of the title problem is constructed using the averaging method of N. S. Bakhvalov. The averaged equation is of an elliptic type; for small stiffness of the hinge, it is singularly perturbed, and for zero stiffness of the hinge, it is of a composite type. For the first boundary-value problem, a solution of the original problem is proved to converge to that of the limiting problem. A situation where natural boundary conditions are specified for the composite equation is treated. It is shown that the solution space of the homogeneous problem is infinite-dimensional. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 168–174, May–June, 1999.     

Identification of viscoelastic characteristics of composite materials on the basis of results of an experimental and theoretical analysis of the dynamic behavior of hemispherical shells

A method is proposed for determining the stiffness and rheological characteristics of composite materials, which is based on minimizing the disagreement between experimental data and results of numerical simulations of deformation of hemispherical shells under explosive loading. The damping characteristics of randomly reinforced polymer materials are analyzed with the use of this method. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 126–133, May–June, 2006.     

Critical phenomena in particle dissolution in the melt during electron-beam surfacing

A model for the electron-beam surfacing process is proposed that takes into account the dissolution of the modifying particles in the melt. Critical conditions are determined for various modes of surfacing resulting in nearly homogeneous or composite coatings. A detailed parametric study of the one-dimensional version of the model is performed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 131–142, January–February, 2007.