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Mathematical & Mechanical Method in Mechanical Engineering. Dr. Wang Xingbo Fall , 2005. Mathematical & Mechanical Method in Mechanical Engineering. Introduction to Tensors. Concept of Tensors Tensor Algebra Tensor Calculus Application of Tensors. Mathematical & Mechanical
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Mathematical & Mechanical Method in Mechanical Engineering Dr. Wang Xingbo Fall,2005
Mathematical & Mechanical Method in Mechanical Engineering Introduction to Tensors • Concept of Tensors • Tensor Algebra • Tensor Calculus • Application of Tensors
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors A coordinate transformation in an n-dimensional space
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors T is a quantity with ns components represented by one of the following three forms
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the first one and transformed by T is called a contravariant tensor of order s.
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the second one and transformed by T is called a covariant tensor of order s.
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the third one and transformed by T is called a mix-variant tensor of order s.
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor product Let U, V be two vector spaces of dimension m, n, Tensor product of U and V is an mn–dimensional vector space W denoted by W=UV. Symbol is used to denote a tensor product
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Symbol is used to denote a tensor product
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Since UV is an mn-dimensional space, it has mn basis vectors. All pairs (i,j) produce exactly mn pairs of (ui,vj) It often uses symbol to denote the basis of W=UV. The elements of W=UV are
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Covariant tensor basis are defined by tensor product of covariant vector basis
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Contravariant tensor basis are defined by tensor product of contravariant vector basis
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Mix-variant tensor basis are defined by tensor product of covariant and contravariant vector basis
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors A vector is a first-order tensor Take s =1, the two forms of components The transformations This is what a contravariant vector or a covariant vector is!
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor contravariant tensor T can be represented by The transformations
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor covariant vector T can be represented by The transformations
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor Mix covariant vector T can be represented by The transformations
Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor Matrix Form
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Sample is a covariant tensor of order 2
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Quantity to illustrate strain in an elastic material is a covariant tensor of order 2 Let A, B be two points in an elastic body and let .
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor After deformation
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Let us change the Cartesian coordinate transformation O toAssume
Mathematical & Mechanical Method in Mechanical Engineering Second order tensor
Mathematical & Mechanical Method in Mechanical Engineering Strain tensor
Mathematical & Mechanical Method in Mechanical Engineering Tensor algebra Addition and Subtract of Two Tensors Contraction of Tensors : Forcing one upper index equal to a lower index and invoking the summation convention A special operation on mix-variant tensors
Mathematical & mechanical Method in Mechanical Engineering Geometric Meanings of Cross Product Contraction of Tensors
Mathematical & mechanical Method in Mechanical Engineering Out Product of Two Tensors The product of two tensors is a tensor whose order is the sum of the orders of the two tensors, and whose components are products of a component of one tensor with any component of the other tensor.
Mathematical & mechanical Method in Mechanical Engineering Out Product of Two Tensors • A=AikEik , B=BlmElm, • Ciklm= AikBlm
Mathematical & mechanical Method in Mechanical Engineering Inner product of Two Tensors Multiplying two tensors and then contracting the product with respect to indices belonging to different factors
Mathematical & mechanical Method in Mechanical Engineering Quotient Law Assume are two arbitrary tensors. IF Then A is a tensor
Mathematical & mechanical Method in Mechanical Engineering Some Useful and Important Tensors • Metric tensors
Mathematical & mechanical Method in Mechanical Engineering Metric tensors In 3-dimensional space
Mathematical & mechanical Method in Mechanical Engineering The Alternating Tensor of Third Order εjkl= 1, if j, k, l cyclic permutation of 1, 2, 3 εjkl= -1, if j, k, l cyclic permutation of 2, 1, 3 εjkl= 0, otherwise.
Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential be a vector field in covariant frame-vectors dA is called absolute differential or covariant differential of vector field A
Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential Absolute differential of A is composed of two parts reflects the relationship of the contravariant components changing with spatial position
Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential reflects that of frame-vectors components changing with spatial position
Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential Let then absolute derivative represented by contravariant components
Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential We also can derive absolute derivative represented by covariant components
Mathematical & Mechanical Method in Mechanical Engineering Absolute Derivatives & Differential the absolute derivative of a vector field can be represented by either contravariant components or covariant components.
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols are vectors , be linear combination of frame-vectors
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols the first kind of Christoffel symbols the second kind of Christoffel symbols
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols Partial Derivative Operator
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols Relationships between and the metric tensor
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols
Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols The Christoffel symbols are only symbols but not components of any tensor though they look like the form of tensor-components