| Coordinate systems. Vector operations. Transformation rules and matrices. Fundamentals of the indicial notations. Summation convention. Kronecker delta, permutation symbol, box product, determinant. Useful relations for the permutation symbol.Tensor algebra. Permutation symbol with useful relations. Scalars, vectors and the tensor of order two as a homogeneous linear mapping. Tensor product. Special tensors: identity tensor, zero tensor, the transpose of a tensor, symmetric and skew tensors. The eigenvalue problem of symmetric tensors. Tensors of higher order.The eigenvalue problem of symmetric tensors. Tensors of higher order. Isotropic tensors.Some elements of the tensor analysis. Gradient, divergence and the divergence theorem.Classification. Configurations. Material and spacial descriptions. Position vectors. Motion law. Displacement vector. Deformation gradient. Inverse deformation gradient. Strain tensors in the initial configurations. Right Cauchy-Green tensor. Green-Lagrange strain tensor.Strain tensors in the current configurations. Left Cauchy-Green tensor. Euler-Almansi strain tensor. Strain tensors in terms of the displacement vector. Relation between the Green-Lagrange and Euler-Almansi strain tensors. Polar decomposition theorem.Strain measures: Stretch ratio, axial strains, angle changes, vectorial and scalar area elements, relation between volume elements. Compatibility conditions.Time dependent tensor fields. Material time derivative. Velocity gradient. Additive resolution of the velocity gradient. Spin tensor, rotation tensor. Time derivative of the deformation gradient.Time derivatives of strain measures (scalar line element, angle of two line elements, area elements, volume elements). Time derivative of a volume integral.Time derivatives of strain tensors. The Jaumann objective time derivative.The linear theory of deformations. The displacement gradient and its additive resolution: strain tensor and rotation tensor. Compatibility of the linearized strain tensor.External and internal forces. Stress tensors (Cauchy and Piola Kirchhoff I and II). Boundary conditions. Principle of mass conservation, The fundamental theorem of dynamics. Equations of motion.Energy theorem. The first and second laws of thermodynamics. Special vector and tensor fields: kinematically and statically admissible vector and tensor fields).Principle of virtual power (work). Principle of complementary virtual power (work).Incremental form of the principle of virtual work in Lagrangian description. Equations of continuum mechanics: the missing equations.Constitutive equations. The linearized theory of continuum mechanics -- elastic bodies. Field equations and boundary conditions. |
