On an inverse problem of bending of a physically nonlinear inhomogeneous plate

An elastic plate with a physically nonlinear inclusion of an arbitrary shape is considered. This plate is subjected to pure bending under the action of transverse forces and bending moments applied at the external boundary of the plate. There are no loads distributed over the surface. The problem of finding external actions that provide a necessary uniform moment state in the inclusion, i.e., prescribed constant moments and curvatures, is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 104–107, September–October, 2007.     

Stability of plates made of a damageable physically nonlinear material

The bifurcation instability problem for rectangular plates made of physically nonlinear materials progressively damaged with increasing load is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 79–88, September 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.     

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.     

含椭圆刚性核有限大复合材料层板弯曲应力分析Bending problem of a finite composite laminated plate with elliptical rigid inclusions

基于经典的复合材料层板理论,将有限大复合材料层板等效成各向异性弹性平板。采用复变函数理论中的Faber级数分析方法,使用最小二乘边界配点法,对含多椭圆刚性核有限大各向异性板弯曲问题进行应力分析,得到了该问题的级数解形式,分析了含椭圆刚性核层板在弯曲载荷下的应力分布,并讨论了形状和结构参数对应力分布的影响。结果表明,本文方法对于分析含多个椭圆形刚性核有限大薄板弯曲应力问题非常有效,该方法具有精度高及计算方便等优点。     

Stability of cylindrical shells made of a damageable physically nonlinear material

The bifurcation-instability problem for cylindrical shells made of physically nonlinear materials progressively damaged with increasing load is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 10, pp. 56–66, October 2006.     

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.     

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.     

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 fibrous material

The structural theory of short-term damage is generalized to the case where the undamaged isotropic matrix of a fibrous composite with transversely isotropic fibers deforms nonlinearly, with microdamages occurring only in the matrix. The basis for this generalization is the stochastic elasticity equations for a fibrous composite with porous matrix whose skeleton deforms nonlinearly. Microvolumes of the matrix meet the Huber-Mises failure criterion. The damaged microvolume balance equation is derived for the physically nonlinear material of the matrix based on the properties of the ultimate microstrength distribution. Together with the equations relating macrostresses and macrostrains of the fibrous composite with porous nonlinear matrix, they constitute a closed-form system. 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. Uniaxial tension curves are plotted for a fibrous composite with linearly hardening matrix.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 88–97, October 2004.     

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α(ζ), β(ζ), φ(ζ), and ψ(ζ) when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func-tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional (3D) elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.     

Stability of plates made of a particulate composite with nonlinear elastic matrix and damaged inclusions

The bifurcation instability problem for rectangular plates made of particulate composites with nonlinear elastic matrix and damaged inclusions is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 79–89, July 2007.     

The Lur'e-Vorovich Method in Mixed Problems of Bending of Cylindrical Bodies

The mixed problem of bending of a transtropic thick plate and a short cylinder is solved using the Lur'e-Vorovich homogeneous solutions. The most typical results from numerical experiments for a wide range of physical and geometrical parameters are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 58–65, August 2005.     

NONLINEAR BENDING OF SIMPLY SUPPORTED RECTANGULAR SANDWICH PLATES

In this paper,fundamental equations and boundary conditions of the nonlinearbending theory for a rectangular sandwicl plate with a soft core are derived by meansof the method of calculus of variations.Then the nonlinear bending for a simplysupported rectangular sandwich plate under the uniform lateral load is investigated byuse of the perturbation method and a quite accurate analytic solution is obtained.     

The axisymmetrical elastic field of an ellipsoidal inclusion with a slipping interface

This paper is concerned with the axisymmetrical elastic fields caused by an ellipsoidal inclusion with a slipping interface which undergoes a uniform eigenstrain. The problem is solved under a revolving ellipsoidal coordinate with the aid of Papkovich-Neuber general dipacement formula. In contrast to the perfectly bonded interface, when the interface between the inclusion and the matrix cannot sustain shear stress, and is free to slip, the solution cannot be expressed in closed form and involves infinite series. Therefore, the results are illustrated by numerical examples.     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES (III)—CANTILEVER RECTANGULAR PLATES

In this paper, applying the method of the reciprocal theorem, we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates, the figures and tables of number value of bending moment and the deflection amplitudes as well.     

Stability of convex shells of revolution made of particulate composites with physically nonlinear matrix and damageable inclusions

The problem of bifurcation instability of shells of revolution made of particulate composites with physically nonlinear matrix and damageable inclusions is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 70–80, June 2008.     

任意边界条件下双模量矩形薄板的弯曲

将双模量板等效为两个各向同性小矩形板组成的层合板,假定该层合板的中性面即为两个小矩形板的交界面。根据中性面上应力为零且薄板全厚度上应力的代数和为零,推导了双模量矩形薄板的中性面位置。本文采用严宗达提出的带补充项的双重正弦傅里叶级数通解,该通解可以适用于任意边界条件的矩形薄板且不需要叠加或者重新构造。联立边界条件和控制方程,求得通解中的待定系数并代入到通解中,即可得到任意边界条件下双模量矩形薄板的弯曲解析解。与有限元结果比较,本文结果符合工程精度要求。     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。