1. science
2. mathematics
3. algebra

An Introduction to
Nonlinear Solid Mechanics
Marino Arroyo & Anna Pandolfi
An Introduction to Nonlinear Solid Mechanics
Doctoral School – Politecnico di Milano
November 2014
Course Outline
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Continuum mechanics explains successfully various phenomena without
describing the complexity of the internal microstructures:
the solid is seen as a macroscopic system.
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Mathematics preliminaries
Kinematics of deformations
Statics
Balance principles
Objectivity
Thermodynamics of materials
Hyperelasticity
Plasticity
Viscosity
Other behaviors
Lecture slides: http://www.stru.polimi.it/people/pandolfi/WebPage/nlsm.plp
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Reference Textbooks
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G. A. Holzapfel, Nonlinear Solid Mechanics,
Wiley, Chichester, 2000
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R. W. Ogden, Nonlinear Elastic Deformations,
Constable Company, London, 1984 (Dover, New York, 1997)
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J. E. Mardsen and T.J.R. Hughes, Mathematical Foundations of Elasticity
Englewood Cliffs, N.J. Prentice-Hall, 1984 (Dover, New York, 1994)
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A. J. M. Spencer,Continuum Mechanics,
Longman, London, 1980 (Dover, New York, 2004)
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P. Chadwick, Continuum Mechanics. Concise Theory and Problems,
Allen & Unwin Ltd, London,1976 (Dover, New York, 1999)
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L. E. Malvern, Introduction to the Mechanics of a Continuous Medium
Prentice-Hall, Englewood Cliffs, New Jersey, 1969
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C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics,
Springer-Verlag, Berlin, 1965 (1992, 2004)
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1. Mathematical preliminaries
Anna Pandolfi
An Introduction to Nonlinear Solid Mechanics
Doctoral School – Politecnico di Milano
January – February 2012
Notation
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Scalars:
Vectors:
2-nd order tensors:
3-rd order tensors:
4-th order tensors:
Dot product:
Vector product:
Dyadic product:
Norm:
Vector or Na bla operator:
Divergence:
Curl:
Laplace operator:
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Use the dummy (repeated) index over 1,2,3 (3D space) to denote summation.
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Orthonormal Basis Vectors
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3D Euclidean space with a right-handed orthonormal system (basis).
Fixed set of three unit vectors
with the property:
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Representation of vectors in index notation:
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Permutation tensor:
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Properties of orthonormal basis vectors:
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Basic Vector Algebra
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Scalar product (contraction of indices):
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jth component of a vector:
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Norm of a vector:
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Vector (or cross) product:
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Triple scalar (or box) product:
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Groups of Linear Transformations
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A group is a set X together with a binary operation
1. The operation is associative
2. The identity element e is defined
3. The inverse element x-1 is defined
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By fixing a basis in
, identify the set of the linear mappings by the set
of all the square n x n matrices. The matrix multiplication can be
assumed as the corresponding binary operation.
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Matrix multiplication represents the composition of all linear mappings
from
to
on the set
, but it fails to be a group since not all the
linear mappings have an inverse. It requires a restriction.
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The General Linear Group GL(n) of dimension n is the group of invertible linear
mappings over
:
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s.t.
Second Order Tensors
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A second order tensor A is linear transformation (linear mapping) that acts on u
to generate v :
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How can a tensor be generated? Through the combination of dyads, i.e. dyadic
(or tensor or outer) products of vectors. A dyadic product results into an
“expansion of indices” (as opposite to inner or dot product, which is a
“contraction of indices”)
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A dyadic W is a linear combination of dyads with scalar coefficients (in general
no tensor can be expressed as a single dyad):
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Any second order tensor A may be expressed as a dyadic formed by the basis
vectors (proved after the example):
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Interpretation of Dyad
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The dyad is a second order tensor which linearly transforms a vector w into a
vector n in the direction of u with the rule:
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Simple example in Cartesian coordinates. Assume:
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Now compute:
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The vector w transforms into a vector
in the direction of u. The figure shows the
special case on unit vectors.
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Cartesian Tensors
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When a tensor A is resolved along orthonormal basis vectors, it is called
Cartesian tensor. In particular, define the unit tensor I:
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A dyad
and a tensor A can be expressed in matrix notation by entering
their nine Cartesian components into a table:
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The Cartesian components of A can be computed by using the dyad definition:
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Using the following steps:
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The expression shows that the vector
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has three components in
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Positive and Negative Definiteness
Trace and Transpose
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The tensor A is said positive semi-definite (negative semi-definite) if for any non
zero vector v:
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The tensor A is said positive definite (negative definite) if for any non zero
vector v:
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The positive or negative definiteness are related to the associate eigenvalue
problem.
Trace of a dyad or of a tensor:
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Transpose AT of a tensor:
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In particular:
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Contraction Products
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Contraction of an index: a familiar example is the scalar product.
Single index contraction product (or dot product) can be defined between two
tensors and a tensor and a vector:
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yielding to a tensor or to a vector, respectively.
Components of a dot product along an orthonormal basis:
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A double index contraction product between two tensors yields a scalar
(example: internal work in WVP):
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Application to the calculation of the norm of a tensor:
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Determinant and inverse of a Tensor
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Determinant of a tensor:
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Properties:
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If the determinant is null, the tensor is said to be singular. For a nonsingular
tensor, a unique inverse A-1 exists, such that:
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Shortly, denote:
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If the unique inverse A-1 does exist, the tensor A is said invertible.
The General Linear Group GL(n) is the group of the invertible linear mappings
over .
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Orthogonal Tensors
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A tensor is said orthogonal if it preserves the length and the angle of vectors:
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Which implies:
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By computing the determinants:
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If Q and R are orthogonal, then also the product QR is orthogonal. It is said that
the set of orthogonal mappings is a subgroup of GL(n) closed under
multiplication, and it is called Orthogonal Group O(n).
The subgroup of the orthogonal group that preserves the orientation, i.e. with
determinant equal to 1, is called Special Orthogonal Group SO(n). The other
subgroup corresponds to reflections.
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Symmetric and Skew Tensors
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Any second order tensor A can always be uniquely decomposed into a
symmetric S and a skew W tensor:
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The meaning and the importance of a symmetric tensor will be discussed later.
Any skew tensor W behaves like a three component vector. By introducing the
axial (dual) vector w of W, for any vector u the following relation holds:
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The components of the skew tensor and of the axial vector are related as:
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Example:
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Projection Tensors
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Projections: given a unit vector e, a vector u decomposes into a vector in the
direction of e and a vector onto the plane normal to e:
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where the projection tensors Pe of order two are introduced.
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Spheric and Deviatoric Tensors
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Any second order tensor A can be decomposed into the sum of a spherical part
and a deviatoric part:
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Deviatoric operator:
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Example: von Mises theory of plasticity where the deviatoric part of the stress is
used
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The spherical part of a second order tensor is also said isotropic. The elements
outside of the principal diagonal of a spherical tensor are null.
The deviatoric part of a second order tensor is also said isochoric. The trace of
a deviatoric tensor is null.
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Higher Order Tensors
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Three-order tensors (27 components):
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The double contraction between a 3-order and a 2-order tensors gives a vector:
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Four-order tensors (81 components):
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The double contraction between a 4-order and a 2-order tensors gives a 2 order
tensor:
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Obtain a four-order tensor by a dyadic product of 2 two-order tensors:
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It is a typical tensor used in constitutive relations, to link a second order tensor
(stress) to another (strain).
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Transpose and Identity 4th Order Tensors
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Unique transpose of a four-order tensor: for all second order tensors B and C
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Two (distinct) identity four-order tensors:
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The deviatoric part of a second order tensor A may be written alternatively by
introducing a four-order projection tensor :
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Eigenvalues of Tensors
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An eigenvalue of a A is a complex number
such that
is not
invertible. The spectrum
is the collection of all eigenvalues of A.
The characteristic polynomial
is invariant under similarity
transformations; the coefficients are also invariants:
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Eigenvalues are the roots of the characteristic polynomial:
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Eigenvalues characterize the physical nature of a tensor and do not depend on
coordinates.
Since
, the complex eigenvalues came in conjugate pairs.
A and AT have the same eigenvalues. If A is invertible, the eigenvalues of A-1
are the reciprocal of the eigenvalues of A.
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Eigenvectors of General Tensors
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A right (or left) eigenvector
associated to a eigenvalue l of a
second order tensor A satisfies the equation:
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A right eigenvector u1 and a left eigenvector v2 are orthogonal if and only if:
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A is said to be diagonalizable if it has n linearly independent eigenvectors. In
such a case it is possible to define a right and a left eigenbasis (dual basis) with
normalized eigenvectors, so that:
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Then A and AT admit the so called spectral decomposition:
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Symmetric Tensors, Spectral Decomposition
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In the case of symmetric tensors, eigenvalues are real numbers, and right and
left eigenvectors coincide. The eigenvectors form an orthonormal basis and the
tensor is diagonalizable.
By using the definition of identity second order tensor in the principal reference
system, write any diagonalizable tensor with distinct eigenvalues as:
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For two equal eigenvalues (e.g. in 2 dimensions) , i.e. l1=l2=l :
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For three equal eigenvalues:
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Transformation Laws for Basis Vectors
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Vector and tensor remain invariant upon a change of basis, but their respective
components depend upon the coordinate system introduced.
Given two sets of basis vectors,
and
, , denote
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Using the dyadic representation by the basis of the tensor Q :
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[Q] is an orthogonal matrix which contains the collection of the components Qij
of the proper orthogonal tensor Q and it is referred to as the transformation
matrix.
In the following, we use [u] to denote the column matrix representation of a
vector and [A] the matrix representation of a tensor.
In different basis the same vector (tensor) is described by different components.
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Vector and Tensor Transformation Laws
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Vector transformation law. Describe the same vector in two different systems:
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Tensor transformation law. Describe the same tensor in two different systems:
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A tensor is said to be isotropic if its components are the same under arbitrary
rotations of the basis vectors.
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Scalar, Vector and Tensor Functions
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Scalar, Vector and Tensor functions: the arguments are scalar, vector, or tensor
variables respectively. The returned values may be any. For vector and tensor
values we assume that the components, in a basis system ei, are in turn
functions of the scalar, vector, or tensor variables:
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First (nth) derivatives wrt a scalar argument (i.e. time t), as derivatives of the
components:
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Derivative of the inverse:
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Gradient of a Scalar valued Tensor Function
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The nonlinear scalar-valued function
returns a scalar for each A. Want to
approximate the nonlinear function F at A with a linear function: use first-order
Taylor’s expansion
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The Landau order symbol o(•) denotes a small error that tends to zero faster
than his argument:
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Second order tensor called gradient (or derivative) of F at A:
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The gradient is derived as the first variation of F.
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Gradient of a Tensor valued Tensor Function
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The nonlinear tensor valued tensor function
returns a tensor for each B.
Want to approximate the nonlinear function A at B by a linear function: use firstorder Taylor’s expansion
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Fourth-order tensor called gradient (or derivative) of function A at B:
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In particular:
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Gradient of a Scalar Field F (x)
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A scalar field F(x), a vector field u(x), or a tensor field A(x) assign a scalar,
vector, or tensor respectively to each material point x over a domain W.
The Taylor’s expansions of a scalar field introduces the gradient:
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Total differential of F:
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Introduce the Vector operator (Nabla operator) :
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Directional Derivative
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Level surface of a scalar field F in x:
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Since gradF is a vector normal to the level
surface, the normal to the surface n is:
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For any vector u forming an angle q with n, the directional derivative (Gâteaux
derivative) at x in the direction of u is the scalar product:
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It is maximum in the direction of n and minimum in the direction of –n.
The normal derivative is the maximum of all the directional derivatives in x for
unit vectors u:
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Alternative Definitions of Directional Derivative
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Gâteaux derivative: for any vector u forming an angle q with n, the directional
derivative at x in the direction of u is the scalar product:
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The directional derivative in x is also the rate of change of F along the straight
line through x in the direction of u:
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Divergence and Curl
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Divergence: scalar field produced by the vector operator dotted in a vector field
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When div u = 0, the field is said solenoidal (or divergence-free)
Curl: vector field produced by the vector operator crossed in a vector field
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When curl u = 0, the field is said irrotational (or conservative, curl-free). It holds:
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A vector field u such that u = grad F is automatically irrotational.
ThusF is called potential of u.
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Gradient of a Vector Field
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The gradient of a smooth vector field u(x) is a second order tensor field:
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in matrix notation:
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Note that:
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Transpose of the gradient:
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Divergence and Gradient of a Tensor Field
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Divergence of a tensor field: vector field produced by the vector operator dotted
in a tensor field
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Alternative definition (differential equilibrium equation)
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The gradient of a smooth tensor field A(x) is a third-order tensor field:
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Note that:
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Laplace operator: the vector operator dotted into itself
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Special Differential Equations and Hessian
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The Laplace operator operated upon a scalar field F yields another scalar field,
as in the Laplace’s equation (Poisson’s equation):
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Hessian operator (gradient of a gradient, leads to a tensor):
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Properties (to remember in the calculation of the tangent stiffness):
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If the vector field is both solenoidal and irrotational, it is said to be harmonic:
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Integral Theorems
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Divergence Theorem: for u and A
smooth fields defined over a 3D
region with volume V and bounding
closed surface S:
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Gauss’ divergence theorem: the first integral is called total (outward normal) flux
of u out of the total boundary surface S enclosing V.
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