A nonlinear composite plate theory

A general nonlinear theory for the dynamics of elastic anisotropic plates undergoing moderate-rotation vibrations is presented. The theory fully accounts for geometric nonlinearities (moderate rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. The theory accounts for transverse shear deformations by using a third-order theory and for extensionality and changes in the configuration due to in-plane and transverse deformations. Five third-order nonlinear partial-differential equations of motion describing the extension-extension-bending-shear-shear vibrations of plates are obtained by an asymptotic analysis, which reveals that laminated plates display linear elastic and nonlinear geometric couplings among all motions.     

A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.     

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.     

Studying the harmonic vibrations of a cylindrical shell made of a nonlinear elastic piezoelectric material

The paper examines the harmonic vibrations of an infinitely long thin cylindrical shell made of a nonlinear elastic piezoceramic material and subjected to periodic electric loading. Amplitude-frequency characteristics are plotted for different amplitudes of the load. Points of these characteristics are analyzed for stability. The transients occurring while harmonic vibrations attain the steady state are studied __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 101–106, April 2008.     

Three-dimensional nonlinear vibrations of composite beams — I. Equations of motion

Newton's second law is used to develop the nonlinear equations describing the extensional-flexural-flexural-torsional vibrations of slewing or rotating metallic and composite beams. Three consecutive Euler angles are used to relate the deformed and undeformed states. Because the twisting-related Euler angle is not an independent Lagrangian coordinate, twisting curvature is used to define the twist angle, and the resulting equations of motion are symmetric and independent of the rotation sequence of the Euler angles. The equations of motion are valid for extensional, inextensional, uniform and nonuniform, metallic and composite beams. The equations contain structural coupling terms and quadratic and cubic nonlinearities due to curvature and inertia. Some comparisons with other derivations are made, and the characteristics of the modeling are addressed. The second part of the paper will present a nonlinear analysis of a symmetric angle-ply graphite-epoxy beam exhibiting bending-twisting coupling and a two-to-one internal resonance.     

Determining the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Shells by the Theory of Deformation Along Paths of Small Curvature

A technique is developed for determining the thermoviscoelastoplastic geometrically nonlinear axisymmetric stress–strain state of laminar shells of revolution under loads that induce meridional stress and torsion. The technique is based on the hypotheses of rectilinear element for the whole stack of layers. The relations of the theory of deformations along paths of small curvature are used as equations of state. The solution is reduced to the numerical integration of a system of ordinary differential equations. The technique is tried out by a test example and illustrated by determining the geometrically nonlinear thermoviscoelastoplastic state of a corrugated shell     

Six-variable geometrical nonlinear laminated theory for large deformation

A six-variable geometrical nonlinear shear deformation laminated theory is presented by which normal stress and strain distribution can be calculated. By considering some affective factors that were neglected under the finite deformation condition, an improved von Karman geometrical nonlinear deformation-strain relation is used for large deformation analysis. After analyzing the bending problem of laminated plates, and comparing it with 3-D elasticity solutions and J. N. Reddy five-variable simple higher-order shear deformation laminated theory, we can conclude that a satisfactory calculation precision has been achieved, which shows that it is especially suitable for the calculation in the condition of large deformation and the laminated thick plate analysis.     

A new higher-order shear deformation theory and refined beam element of composite laminates

A new higher-order shear deformation theory based on global-local superposition technique is developed. The theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. The global displacement components are of the Reddy theory and local components are of the internal first to third-order terms in each layer. A two-node beam element based on this theory is proposed. The solutions are compared with 3D-elasticity solutions. Numerical results show that present beam element has higher computational efficiency and higher accuracy.The project supported by the National Natural Science Foundation of China (10172023)     

Features of nonlinear deformation and critical states of shell systems with geometrical imperfections

Results from theoretical and experimental investigations into the nonlinear deformation (geometrical nonlinearity, plastic deformation, creep) and critical states (limit loads, buckling) of shell-frame systems with geometric imperfections are analyzed. The presence of prestresses is allowed. Diverse effects of various geometrical imperfections and plastic deformation for different load histories are studied. New qualitative effects of the mutual influence of these factors are established. Relevant experimental results are outlined __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 12, pp. 3–47, December 2006.     

A large deformation analysis for nonlinear elastic problems by hypoelastic theory

In this paper, large deformation problems for nonlinear elasticity are studied on the basis of hypoelastic theory. The expression of Cauchy elasticity is derived in the form of hypoclasticity. This makes it possible to solve large deformation for nonlinear elastic problems by the hypoelastic method. The variational principle of hypoclasticity and the Ritz method are used to obtain the numerical solution of a rectangular rubber membrane under uniaxial stretch.     

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 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.     

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.     

Modeling cylindrical waves in nonlinear elastic composites

A procedure of deriving nonlinear wave equations that describe the propagation and interaction of hyperelastic cylindrical waves in composite materials modeled by a mixture with two elastic constituents is outlined. Nonlinearity is introduced by metric coefficients, Cauchy-Green strain tensor, and Murnaghan potential. It is the quadratic nonlinearity of all governing relations. For a configuration (state) dependent on the radial coordinate and independent of the angular and axial coordinates, quadratically nonlinear wave equations for stresses are derived and a relationship between the components of the stress tensor and partial strain gradient is established. Four combinations of physical and geometrical nonlinearities in systems of wave equations are examined. Nonlinear wave equations are explicitly written for three of the combinations __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 63–72, June 2007.     

A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells

A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain-displacement relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Kármán type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Kármán type for vibration amplitudes of about two times the shell thickness for the studied case.     

参数和直接激励下非线性压电俘能器性能分析的多尺度法

考虑几何非线性、阻尼非线性和梁的轴向不可伸长条件,利用Hamilton变分原理,建立了参数激励和直接激励下压电俘能器的非线性力电耦合的运动微分方程;利用Galerkin法,将所建立的动力学偏微分方程降阶为力电耦合的Mathieu-Duffing型方程;采用多尺度法获得了梁的位移和输出电压的解析表达式,给出了解的稳定性条件;利用解析表达式研究了单独参数激励以及参数激励和直接激励共同作用下阻尼系数对压电俘能器性能的影响。结果表明,在参数激励情况下,线性阻尼会显著影响超临界分岔点的位置,非线性二次阻尼不会影响超临界分岔点的位置。参数激励和直接激励的结合可以作为提升压电能量俘获器性能的解决方案。     

Stability of shells of revolution made of a particulate composite with nonlinear elastic inclusions and damageable matrix

The bifurcation instability problem for shells of revolution made of particulate composites with nonlinear elastic inclusions and damageable matrix is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 55–65, November 2007.     

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

A self-consistent solution for nonlinear theory of the positive column in a magnetic field

Only the electron and ion gases were taken into account in all previous theories of the positive column of intermediately-low-pressure arc discharge with or without the longitudinal magnetic field, while the motion of neutral gas was neglected. In 1982, the authors^[1] presented a nonlinear theory of a positive column which indicated that the rotating velocities of neutral gas and ion gas were nearly equal, and the motion of neutral gas could not be ignored. They further discussed the problem of validity of Bohm's criterion. However, some of the parameters with which the computation was worked out in Ref. [1] were not correlated to the initial discharge parameters. In the present paper, two integral relations are supplemented, so that a complete mathematical formation of the problem is given. A convergent numerical solution is obtained by iteration and the solution of Ref. [1] turns out to be the first iteration approximation. It is shown that both functions and parameters obtained by self-consistent solution differ significantly from those obtained in the first iteration approximation. According to this paper the computation can be conducted when the initial discharge parameters are given, so this method could have certain practical applications.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.