Multiscale Biomechanics

Multiscale Biomechanics provides new insights on multiscale static and dynamic behavior of both soft and hard biological tissues, including bone, the intervertebral disk, biological membranes and tendons. The physiological aspects of bones and biological membranes are introduced, along with micromechanical models used to compute mechanical response. A modern account of continuum mechanics of growth and remodeling, generalized continuum models to capture internal lengths scales, and dedicated homogenization methods are provided to help the reader with the necessary theoretical foundations. Topics discussed include multiscale methods for fibrous media based on discrete homogenization, generalized continua constitutive models for bone, and a presentation of recent theoretical and numerical advances.

In addition, a refresher on continuum mechanics and more advanced background related to differential geometry, configurational mechanics, mechanics of growth, thermodynamics of open systems and homogenization methods is given in separate chapters. Numerical aspects are treated in detail, and simulations are presented to illustrate models.

This book is intended for graduate students and researchers in biomechanics interested in the latest research developments, as well as those who wish to gain insight into the field of biomechanics.

• Provides a clear exposition of multiscale methods for fibrous media based on discrete homogenization and the consideration of generalized continua constitutive models for bone
• Presents recent theoretical and numerical advances for bone remodeling and growth
• Includes the necessary theoretical background that is exposed in a clear and self-contained manner
• Covers continuum mechanics and more advanced background related to differential geometry, configurational mechanics, mechanics of growth, thermodynamics of open systems and homogenization methods

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 3, 2018
• ISBN: 9780081021156

Book preview

proposed.

Part 1

Theoretical Basis: Continuum Mechanics, Homogenization Methods, Thermodynamics of Growing Solid Bodies

1

Tensor Calculus

Jean-François Ganghoffer

Abstract

The concepts of tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word tensor itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor (Note 1) The contemporary usage was brought in by Voigt in 1898. Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892. It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (translated into Methods of absolute differential calculus and their applications).

Keywords

Covariant and contravariant tensors; Curvilinear coordinates; Eigenvalues; Eigenvectors; Euclidean tensors; Frechet and Gateaux derivatives; Orthogonal tensors; Riesz representation theorem; Tensor algebra

1.1 A short historical vignette

The concepts of tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word tensor itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor (Note 1) The contemporary usage was brought in by Voigt in 1898. Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892. It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita’s 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (translated into Methods of absolute differential calculus and their applications).

In the 20th Century, the subject came to be known as tensor analysis, and was widely developed due to Einstein’s theory of general relativity, around 1915. General relativity is indeed formulated completely in the language of tensors, a field Einstein had learned about with great difficulty from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. In mathematics, the exterior algebra of Hermann Grassmann (middle of the nineteenth century), and the work of Élie Cartan on differential forms are more recent developments of the concept of tensor.

Tensor fields are extensively used in field theories including electromagnetism, continuum mechanics and general relativity, as well as in numerous applications in physical sciences and engineering. A tensor is a generalization of a scalar (a pure number representing the value of some physical quantity) and a vector (a geometrical arrow in space), and a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.

The main idea behind the concept of tensor is to delineate what is intrinsic (independent from a metric or a basis). Intrinsic means independent from any observer, thus from any basis or scalar product. Behind this stands the general notion of covariance: equations of physics have to be independent from any observer, thus should take the same mathematical form in all coordinate systems, which require the use of tensors. The strong form of covariance is invariance, which only hold for scalar valued quantities (they remain the same whatever the coordinate system), whereas vectors and tensors transform in a covariant manner under a change of coordinates (they follow the rule of matrix transformation you already know, but in an extended form).

In order to explain the need for the concept of tensor, while a vector is determined by 3 components, there are quantities requiring more than 3 components to be completely characterized. To be specific, consider a unitary cubic volume element submitted to forces (weight, contact forces) so that internal forces are generated (Figure 1.1).

Figure 1.1 Cubic material element subjected to contact forces. For a color version of this figure, see www.iste.co.uk/ganghoffer/biomechanics.zip

be the face of the cube orthogonal to the basis vector ei of the same index; the face Si is submitted to a force (vector) Fi with a priori an arbitrary orientation: it can be accordingly decomposed along the three basis vectors

are the components of the stress tensor; they completely specify the internal forces acting at each point. Since they are a priori independent, it means that the concept of vector is here clearly inadequate.

Areas of concern in mathematics: multilinear algebra, differential geometry. A vector is a fairly general object, it belongs to a vector space; by extension, one shall introduce the notion of tensor field. Representation of multilinear applications: in this sense, a tensor is intrinsic!

1.1.1 Differencebetweenmatrixandtensor

A matrix is a mere table of numbers, whereas a tensor is an abstract object whose components are changing when the basis is changing, thus they shall satisfy some consistency.

In physics and engineering, tensors are used to describe various physical quantities such as electrical and magnetic fields, many physical properties (direction dependent) such as permittivity, conductivity, and general relativity (space-time as a 4D curved manifold).

In differential geometry, tensors are used to define on a differentiable manifold the geometrical notions of distance, angle, volume. This is done in metric spaces via the choice of a metric, elaborated as a scalar product of vectors in the tangent space at each point.

1.2 Vector spaces

Definition

Examples of free vectors: x, y.

Operations of vector addition: commutative (x + y = y + x), associative (x + (y + z) = (x + y) + z), existence of nil vector (x + 0 + x) and opposite vector (x + (−x) = 0).

Multiplication by a scalar α∈R: 1x = x, associative α(βx) = (αβ)x. Distributive for addition of scalars: (α + β)x = αx + βx; distributive for vector addition: α(x + y) = αx + αy.

Any vector can be expressed using the basis vectors; the number n of those vectors dictates the dimension of the vector space E, denoted En.

Note

Generalization

Any set E of elements x, y,… endowed with two laws ‘addition’ and ‘external multiplication’ on a commutative field K having these properties has the status of a vector space; x, y,… are called vectors (they may be functions or more general objects).

For isomorphisms of vector spaces let us consider circular uniform motion as an example (Figure 1.2).

Figure 1.2 Vectors representing different physical quantities at the same point. For a color version of this figure, see www.iste.co.uk/ganghoffer/biomechanics.zip

Figure 1.2 pictures different vectors, OM distance, V velocity, F force, γ acceleration. These define quantities belonging to vector spaces of a different physical nature (dimensions are different), which are nevertheless represented in the same geometrical space. It is sufficient for this purpose to fix scales allowing interpretation of the norms: scale of distances, velocities, forces, accelerations. This is possible since all those vectors have the same dimension and obey the same calculus rules: there is an isomorphism between the vector spaces they belong to, which allows each of them to be represented in either space. This can be done e.g. in R³.

Note

Einstein notation

, i,j = 1,…,n, with i row index and j column index.

Example

, i, j, k∈1,…,n. Since there is summation for each occurrence of the index in a lower (subscript) and upper position (superscript) in any monomial, the summation symbol can be removed: the summation integer k is mute, so that it can be replaced by any letter (similar to integration variable):

Example

; we shall then write

so that each repeated mute index is not used twice. In the geometrical space R³, all indices vary from 1 to 3.

1.3 Covariant and contravariant tensors

For a given basis e1, e2,…,en of En, denoted collectively {ei}, any vector V can be represented as V = Viei; quantities Vi are the components of V in this basis. For another basis E1, E2,…, En, denoted {E. This can be represented in matrix form as

, denoting a general positioning of indices for the summation to make a correspondence with matrix product.

Remark

In continuum mechanics, one very often considers change of bases given by an orthogonal transformation, so that

In which the scalars (qji): = cos(ej,EI) define the components of an orthogonal transformation Q = [qij]. The ith column of Q is formed by the components of EI in the initial basis.

For a vector V, we have the correspondence V = Viei =VIEI,