Short-term microdamage of a physically nonlinear fibrous material under simultaneous normal and tangential loads

The structural theory of short-term damage is generalized to the case where the undamaged isotropic matrix of a fibrous composite with transversely isotropic reinforcement deforms nonlinearly under loads that induce a combined stress state, microdamages occurring in the matrix alone. The basis for this generalization is the stochastic elasticity equations for a fibrous composite with porous matrix whose skeleton deforms nonlinearly. The Huber-Mises failure criterion is used to describe the damage of microvolumes in the matrix. The damaged microvolume balance equation is derived for the physically nonlinear material of the matrix based on the properties of the distribution function for the statistically homogeneous random field of ultimate microstrength. Together with the macrostress-macrostrain relationship, they constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the dependences of macrostresses and microdamages on macrostrains are proposed. Stress-strain curves for a composite with a linearly hardened matrix under simultaneous normal and tangential loads are plotted. The effect of the volume fraction of reinforcement and tangential load on the curves is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 48–59, March 2007.     

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 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     

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.     

Deformation of Particulate Composite with Physically Nonlinear Inclusions and Microdamageable Matrix

The structural theory of short-term damage is generalized to the case where the matrix of a particulate composite has microdamages and the inclusions deform nonlinearly. The basis for this generalization is the stochastic elasticity equations of a porous-matrix particle-reinforced composite. Microvolumes of the matrix meet the Huber-Mises failure criterion. A balance equation for damaged microvolumes is derived. The balance equation and the equations relating macrostresses and macrostrains of a particulate composite with porous matrix and physically nonlinear inclusions 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 the material of inclusions is linearly hardening__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 3–11, February 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.     

Influence of the Physical Nonlinearity of the Matrix on the Deformation of a Particulate Composite with Microdamageable Inclusions

The structural theory of short-term damage is generalized to particulate composites with nonlinearly elastic matrix and microdamageable inclusions. The basis for this generalization is the stochastic elasticity equations for a particulate composite with porous inclusions. Microvolumes of the material meet the Huber-Mises failure criterion. The damaged-microvolume balance equation and the equations relating macrostresses and macrostrains of a particulate composite with porous inclusions and physically nonlinear matrix constitute a closed-form system. This system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for computing the microdamage-macrostrain relationships and deformation curves are proposed. Uniaxial tension curves are plotted for a particulate composite with linearly hardening matrix__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 3–11, April 2005.     

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.     

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.     

Short-term microdamage of a physically nonlinear laminate under simultaneous normal and tangential loads

The structural theory of short-term damage is generalized to the case where undamaged components of an N-component laminate deform nonlinearly under loads that induce a combined stress state. The basis for this generalization is the stochastic elasticity equations for an N-component laminate with porous components whose skeleton deforms nonlinearly. The Huber-Mises failure criterion is used to describe the damage of microvolumes in the composite. The damaged microvolume balance equation is derived for the physically nonlinear materials of the composite components. Together with the macrostress-macrostrain relationship, they constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage. For a two-component laminate, algorithms for calculating the microdamage-macrostrain relationship and plotting stress-strain curves are proposed. Stress-strain curves are also plotted for the case where microdamages occur in the linearly hardening component and do not in the linear elastic component under simultaneous normal and tangential loads. The effect of the volume fraction of reinforcement and tangential load on the curves is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 62–72, April 2007.     

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 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.     

Microdamage of a nonlinear elastic material in combined stress state

The structural theory of short-term damage is used to study the coupled processes of deformation and microdamage of a physically nonlinear material in a combined stress state. The basis for the analysis is the stochastic elasticity equations for a physically nonlinear porous medium. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. The balance equation for damaged microvolumes is derived and added to the macrostress-macrostrain relations to produce a closed-form system of equations. It describes the coupled processes of nonlinear deformation and microdamage of the porous material. Algorithms are developed for calculating the dependence of microdamage on macrostresses and macrostrains and plotting stress-strain curves for a homogeneous material under either biaxial normal loading or combined normal and tangential loading. The plots are analyzed depending on the type of stress state __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 30–39, November 2006.     

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     

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.     

Mesomechanics of deformation and short-term damage of linear elastic homogeneous and composite materials

The structural theory of microdamage of homogeneous and composite materials is generalized. The theory is based on the equations and methods of the mechanics of microinhomogeneous bodies with stochastic structure. A single microdamage is modeled by a quasispherical pore empty or filled with particles of a damaged material. The accumulation of microdamages under increasing loading is modeled as increasing porosity. The damage within a single microvolume is governed by the Huber-Mises or Schleicher-Nadai failure criterion. The ultimate strength is assumed to be a random function of coordinates with power-law or Weibull one-point distribution. The stress-strain state and effective elastic properties of a composite with microdamaged components are determined using the stochastic equations of elasticity. The equations of deformation and microdamage and the porosity balance equation constitute a closed-form system of equations. The solution is found iteratively using conditional moments. The effect of temperature on the coupled processes of deformation and microdamage is taken into account. Algorithms for plotting the dependences of microdamage and macrostresses on macrostrains for composites of different structure are developed. The effect of temperature and strength of damaged material on the stress-strain and microdamage curves is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 3–42, June 2007.     

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.     

Micromechanics of Short-Term Thermal Damageability

The structural theory of microdamageability of a homogeneous material is generalized to the case of a thermal action. The theory is based on the stochastic thermoelastic equations of a medium with micropores, hollow or filled with particles of a damaged material. This medium models a material with dispersed microdamages. The Schleicher–Nadai fracture criterion is used as the condition of origin of a micropore in a microvolume of an undestroyed material. It is assumed that the particles of the damaged material in the micropores do not resist shear and triaxial tension and behave as the undamaged material under triaxial compression. The porosity balance equation is corrected for the thermal component and together with the relations between macrostresses, macrostrains, and temperature forms a closed system describing the concurrent action of deformation and microdamage. Nonlinear stress–strain diagrams and dependences of microdamage on macrostrain and temperature are constructed     

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     

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