Mechanics of Solids and Shells: Theories and Approximations
Mechanics of Solids and Shells: Theories and Approximations
复制标题
固体和壳力学：理论和近似
DOI：
10.1115/1.1584415
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
J. Petrolito
中科院分区：
文献类型：
--
作者：
G. Wempner;D. Talaslidis;J. Petrolito
目的和范围。机械概念和数学表示。索引符号。系统。总结大会。索引的位置。向量表示法。克罗内克符号。排列的象征。对称和不对称系统。偏导数的缩写。术语。特定的符号。矢量，张量和曲线坐标。曲线坐标，基向量和度量张量。基向量的乘积。向量的分量。表面和体积元素。向量的导数。张量和不变性。张量有关。协变导数。从笛卡尔坐标到曲线坐标的变换。积分变换。连续介质的变形概念。变形介质的几何学。体积和表面的膨胀。与变形系统相关的向量和张量。小区域运动的性质。压力。应变分量的变换。主要菌株。最大剪切应变。主菌株和主方向的测定。极限剪切应变的测定。工程应变张量。应变不变量与体积应变。将运动分解为旋转和变形。工程应变的物理分量。Strain-Displacement关系。应变分量的兼容性。应变和旋转的速率和增量。欧拉应变速率。应变偏量。小应变近似。小应变和中等旋转的近似。小应变和小旋转的近似。应力应力矢量。夫妇的压力。作用于一个无限小的元素。运动方程。应力和内功的张量和不变形式。应力-物理基础的转化。应力态的性质。静水压力。应力偏量。运动方程的可选形式。小应变的意义。中等旋转近似。小应变和小旋转线性理论的近似。例：梁的屈曲。材料行为导论。一般注意事项。热力学原理。过度的熵。热量流动。熵、熵通量和熵产生。内力的功。第一定律和第二定律的不同形式。把的原则。简单测试的观察结果。弹性。无弹力。线弹性材料。单向、正交异性、横向各向同性和各向同性胡克材料。热传导。胡肯材料中的热传导。各向同性弹性系数。能量势的可选形式。平面应力与平面应变的胡克行为。圣维南原则的正当性。屈服条件。各向同性材料的屈服条件。Tresca屈服条件。冯米塞斯屈服准则。塑料的行为。增量应力-应变关系。流动条件的几何解释。热力学解释。弹塑性变形的正切模量。圣维南、列维、普朗特和罗伊斯的方程。亨基应力-应变关系。无屈服条件的塑性内源性理论。理想可塑性的内慢性形式。粘性行为。牛顿流体。线性粘弹性。各向同性线性粘弹性。功与能导论原理。历史的言论。术语。功，动能，和傅里叶不等式。虚功原理。保守力和势能。静止势能原理。补充能量。最小势能原理。结构稳定性。临界负载稳定性。临界负荷附近的平衡状态。小缺陷对屈曲载荷的影响。连续体虚功原理。连续体的平稳势原理。平稳势原理的推广。一般功能和补充部分。静止互补势原理。互补势的极值性质。Hellinger-Reissner泛函与平稳定理。连续体总结的功能和平稳准则。位移Castigliano定理的推广。非弹性的变分公式。各向同性弹性和粘弹性线性理论简介。线性理论的应用和局限性。线性理论的运动学方程。线性运动方程。线性弹性。线性弹性的边值问题。运动学公式。通过位移解。用应力表示。平面应变与平面应力。艾里应力函数。板上圆孔处的应力集中。复变通解。细长杆的简单弯曲。圆柱杆的扭力。线性粘弹性。运动学公式。准静态问题与变量分离。关于位移的准静态问题。关于应力的准静态问题。拉普拉斯变换与弹性问题的对应关系。曲面微分几何导论。曲面的基向量和度量张量。基向量的乘积。基向量的导数。三维空间的度量张量。基本形式。曲率和扭转。体积和面积差。向量，导数和协变导数。表面张量。曲面的格林定理。高斯和科达齐方程。壳理论导论。历史的角度来看。壳理论的本质。当前处理的范围。运动学。紧张和压力。平衡。互补潜力。物理解释。膜理论。小应变的近似。瘦的含义。具有横向剪切应变的胡克壳理论。Kirchhoff-Love约束下的运动学理论。压力和应变。平衡。相容性方程、应力函数和静力几何类比。Hookean壳的本构方程。薄胡克壳的本构方程。内在基希霍夫-爱理论。Kirchhoff-Love壳的塑性。Strain-Displacement方程。小应变和中等旋转的近似。浅壳理论。Refinements-Limitations-References。近似概念介绍。替代逼近方法。简要回顾。有限差分的概念。泛函解的平稳性与近似形式。基于泛函平稳性的节点逼近。具有连续导数的高阶逼近。有限元近似的物理和数学意义。通过潜在收敛逼近。有效的近似，过大的刚度，和一些补救措施。修正的势收敛和效率逼近。不连续位移下的非协调元逼近。壳的有限元基本特征。关于元素估算的补充说明。非线性路径的逼近。引用
Introduction Purpose and Scope. Mechanical Concepts and Mathematical Representations. Index Notation. Systems. Summation Convention. Position of Indices. Vector Notation. Kronecker Delta. Permutation Symbol. Symmetrical and Antisymmetrical Systems. Abbreviation for Partial Derivatives. Terminology. Specific Notations. Vectors, Tensors, and Curvilinear Coordinates Introduction. Curvilinear Coordinates, Base Vectors, and Metric Tensor. Products of Base Vectors. Components of Vectors. Surface and Volume Elements. Derivatives of Vectors. Tensors and Invariance. Associated Tensors. Covariant Derivative. Transformation from Cartesian to Curvilinear Coordinates. Integral Transformations. Deformation Concept of a Continuous Medium. Geometry of the Deformed Medium. Dilation of Volume and Surface. Vectors and Tensors Associated with the Deformed System. Nature of Motion in Small Regions. Strain. Transformation of Strain Components. Principal Strains. Maximum Shear Strain. Determination of Principal Strains and Principal Directions. Determination of Extremal Shear Strain. Engineering Strain Tensor. Strain Invariants and Volumetric Strain. Decomposition of Motion into Rotation and Deformation. Physical Components of the Engineering Strain. Strain-Displacement Relations. Compatibility of Strain Components. Rates and Increments of Strain and Rotation. Eulerian Strain Rate. Strain Deviator. Approximation of Small Strain. Approximations of Small Strain and Moderate Rotation. Approximations of Small Strain and Small Rotation. Stress Stress Vector. Couple Stress. Actions upon an Infinitesimal Element. Equations of Motion. Tensorial and Invariant Forms of Stress and Internal Work. Transformation of Stress-Physical Basis. Properties of a Stressed State. Hydrostatic Stress. Stress Deviator. Alternative Forms of the Equations of Motion. Significance of Small Strain. Approximation of Moderate Rotations. Approximations of Small Strains and Mall Rotations Linear Theory. Example: Buckling of a Beam. Behavior of Materials Introduction. General Considerations. Thermodynamic Principles. Excessive Entropy. Heat Flow. Entropy, Entropy Flux, and Entropy Production. Work of Internal Forces. Alternative Forms of the First and Second Laws. Saint-Venant's Principle. Observations of Simple Tests. Elasticity. Inelasticity. Linearly Elastic Material. Monotropic, Orthotropic, Transversely Isotropic , and Isotropic Hookean Material. Heat Conduction. Heat Conduction in the Hookean Material. Coefficients of Isotropic Elasticity. Alternative Forms of the Energy Potentials. Hookean Behavior in Plane-Stress and Plane-Strain. Justification of Saint-Venant's Principle. Yield Condition. Yield Condition for Isotropic Materials. Tresca Yield Condition. von Mises Yield Criterion. Plastic Behavior. Incremental Stress-Strain Relations. Geometrical Interpretation of the Flow Condition. Thermodynamic Interpretation. Tangent Modulus of Elasto-plastic Deformations. The Equations of Saint-Venant, Levy, Prandtl, and Reuss. Hencky Stress-Strain Relations. Plasticity without a Yield Condition Endochronic Theory. An Endochronic Form of Ideal Plasticity. Viscous Behavior. Newtonian Fluid. Linear Viscoelasticity. Isotropic Linear Viscoelasticity. Principles of Work and Energy Introduction. Historical Remarks. Terminology. Work, Kinetic Energy, and Fourier's Inequality. The Principle of Virtual Work. Conservative Forces and Potential Energy. Principle of Stationary Potential Energy. Complementary Energy. Principle of Minimum Potential Energy. Structural Stability. Stability at the Critical Load. Equilibrium States near the Critical Load. Effect of Small Imperfections upon the Buckling Load. Principle of Virtual Work Applied to a Continuous Body. Principle of Stationary Potential Applied to a Continuous Body. Generalization of the Principle of Stationary Potential. General Functional and Complementary Parts. Principle of Stationary Complementary Potential. Extremal Properties of the Complementary Potentials. Functionals and Stationary Theorem of Hellinger-Reissner. Functionals and Stationary Criteria for the Continuous Body Summary. Generalization of Castigliano's Theorem on Displacement. Variational Formulations of Inelasticity. Linear Theories of Isotropic Elasticity and Viscoelasticity Introduction. Uses and Limitations of the Linear Theories. Kinematic Equations of a Linear Theory. Linear Equations of Motion. Linear Elasticity. The Boundary-Value Problems of Linear Elasticity. Kinematic Formulation. Solutions via Displacements. Formulation in Terms of Stresses. Plane Strain and Plane Stress. Airy Stress Function. Stress Concentration at a Circular Hole in a Plate. General Solution by Complex Variables. Simple Bending of a Slender Rod. Torsion of a Cylindrical Bar. Linear Viscoelasticity. Kinematic Formulation. Quasistatic Problems and Separation of Variables. Quasistatic Problems in Terms of Displacements. Quasistatic Problems in Terms of Stresses. Laplace Transforms and Correspondence with Elastic Problems. Differential Geometry of a Surface Introduction. Base Vectors and Metric Tensors of the Surface. Products of the Base Vectors. Derivatives of the Base Vectors. Metric Tensor of the Three-Dimensional Space. Fundamental Forms. Curvature and Torsion. Volume and Area Differentials. Vectors, Derivatives, and Covariant Derivatives. Surface Tensors. Green's Theorem for a Surface. Equations of Gauss and Codazzi. Theory of Shells Introduction. Historical Perspective. The Essence of Shell Theory. Scope of the Current Treatment. Kinematics. Strains and Stresses. Equilibrium. Complementary Potentials. Physical Interpretations. Theory of Membranes. Approximations of Small Strain. The Meaning of Thin. Theory of Hookean Shells with Trnsverse Shear Strain. Theories under the Kirchhoff-Love Constraint Kinematics. Stresses and Strains. Equilibrium. Compatibility Equations, Stress Functions, and the Static-Geometric Analoty. Constitutive Equations of the Hookean Shell. Constitutive Equations of the Thin Hookean Shell. Intrinsic Kirchhoff-Love Theories. Plasticity of the Kirchhoff-Love Shell. Strain-Displacement Equations. Approximation of Small Strains and Moderate Rotations. Theory of Shallow Shells. Refinements-Limitations-References. Concepts of Approximation Introduction. Alternative Means of Approximation. Brief Retrospection. Concept of Finite Differences. Stationarity of Functionals Solutions and Forms of Approximation. Nodal Approximations via the Stationarity of a Functional. Higher-Order Approximations with Continuous Derivatives. Approximation by Finite Elements Physical and Mathematical Implications. Approximation via the Potential Convergence. Valid Approximations, Excessive Stiffness, and Some Cures. Approximation via the Modified Potential Convergence and Efficiency. Nonconforming Elements Approximations with Discontinuous Displcements. Finite Elements of Shells Basic Features. Supplementary Remarks on Elemental Approsimations. Approximation of Nonlinear Paths. References