Strain History and Polar Decomposition

The effect of two consecutive strains (only two states enter into the calculation of a strain, the states before and after, independently of the actual strain path) can be calculated by premultiplying the transformation matrix of the first strain (its stretch tensor) with that of the second. Unless the two strains are coaxial (their principal directions coincide), however, the resulting cumulative transformation matrix represents not only a strain but also a rigid-body rotation; in that case the matrix is asymmetric. The method of polar decomposition allows one to interpret the combined transformation as if it had come about either by a strain followed by a rotation (right polar decomposition) or by a rotation followed by a strain (left polar decomposition). Let 𝔸 and 𝔹 be two stretch tensors, or transformation matrices, representing each a strain without rotation; and let the strain 𝔹 follow the strain 𝔸. Then the combined transformation matrix 𝔽 is: . . . 𝔹𝔸 = 𝔽 = ℝ𝕌= 𝕍ℝ, (8.1) . . . where 𝔽 results from premultiplication of the earlier stretch 𝔸 with the later 𝔹, where ℝ𝕌 is the “right” and 𝕍ℝ the “left” decomposition of 𝔽, where 𝕌 and 𝕍 are two distinct stretch tensors, and where ℝ is the transformation matrix for a rotation (elements of rotation matrices are indicated by the symbol aij elsewhere in this book). 𝔽 is asymmetric and ℝ differs from the identity matrix (δij) except when 𝔸 and 𝔹 are coaxial. 𝕌 and 𝕍 have the same principal stretches and differ by orientation only. In Problems 120 to 122, false approaches in the search for an appropriate decomposition of an asymmetric transformation were recognized by yielding impossible values for a rotation. Application of eq. (8.1) makes such a trial-and-error approach unnecessary.

Infinitesimal Strain

An elastic material responds to a stress by a change of volume and shape, or strain, which stays constant as long as the stress is maintained. Materials for which strains are completely reversible and proportional to the stresses that cause them are called ideally elastic and are said to follow Hooke’s law. Many actual materials are nearly ideally elastic as long as the stress-induced strains are small. Strain in such materials is the usual means of observing stress, which itself is an abstraction and not directly observable. Strain, as treated in continuum mechanics, is also an abstraction, but one that more closely approaches observable reality. The element of abstraction comes from treating the deformed body as a continuum, with the implication that neighboring material points in an undeformed body remain arbitrarily close neighbors after deformation. Let one point on a stretchable material line (imagine a rubber band) be held in place at the origin of a one-dimensional coordinate system and stretched, throughout but not necessarily uniformly (the rubber band may vary in thickness), by pulling on its free end with the position Δ x. Let the end, as a consequence of the stretching, be moved by the displacement Δ u. Then any original length element Δ x of the line will be changed to a new length Δx+Δu, say, the particular segment starting at the points P before and P′ after the deformation.

Vectors

The reader, even if familiar with vectors, will find it useful to work through this chapter because it introduces notation that will be used throughout this book. We will take vectors to be entities that possess magnitude, orientation, and sense in three-dimensional space. Graphically, we will represent them as arrows with the sense from tail to head, magnitude proportional to the length, and orientation indicated by the angles they form with a given set of reference directions. Two different kinds of symbol will be used to designate vectors algebraically, boldface letters (and the boldface number zero for a vector of zero magnitude), and subscripted letters to be introduced later. The first problems deal with simple vector geometry and its algebraic representation. Multiplying a vector by a scalar affects only its magnitude (length) without changing its direction. Problem 1. State the necessary and sufficient conditions for the three vectors A, B, and C to form a triangle. (Problems 1–9, 12–14, 19–23, and 25 from Sokolnikoff & Redheffer, 1958.) Problem 2. Given the sum S = A + B and the difference D = A – B, find A and B in terms of S and D (a) graphically and (b) algebraically. Problem 3. (a) State the unit vector a with the same direction as a nonzero vector A. (b) Let two nonzero vectors A and B issue from the same point, forming an angle between them; using the result of (a), find a vector that bisects this angle. Problem 4. Using vector methods, show that a line from one of the vertices of a parallelogram to the midpoint of one of the nonadjacent sides trisects one of the diagonals. Two vectors are said to form with each other two distinct products: a scalar, the dot product, and a vector, the cross product.

Effects of Stress

The simplest relationship between stress and strain is Hooke’s law, describing the linear elastic response of solids to stress. Elastic strain (almost in all cases small) is proportional to the applied stress, with one proportionality factor expressing the relationship between normal, and another that between tangential stress and strain. An ideally elastic strain is completely reversed upon removal of the stress that has caused it. Most materials obey Hooke’s law somewhat imperfectly, and that only up to a critical yield stress beyond which they begin to flow and to acquire, in addition to the elastic strain, a permanent strain that does not revert upon stress release. Hooke’s law in this form is applied to materials that are elastically isotropic, or can be assumed to be approximately so. Crystals, however, never are elastically isotropic, nor are crystalline materials consisting of constituent grains with a distribution of crystallographic orientations that departs from being uniform. The response of a crystal to a stress (at a level below the yield stress) consists of a strain determined by a matter tensor of the fourth rank, the compliance tensor s i j k l : . . . ɛij = s i j k l σkl, (7.1) . . . the 81 components of which are constants. Any tensor that describes the linear relationship between two tensors of the second rank is necessarily of the fourth rank, and like other tensors of the fourth rank, the compliance tensor can be referred to a new set of reference coordinates by means of a rotation matrix aij: s i j k l = aimajnakoalp smnop. (7.2) . . . The components of the compliance tensor are highly redundant, first because both the stress and the strain tensors are symmetric, and second because the tensor itself is symmetric. The number of independent components for crystals of the lowest, triclinic (both classes) symmetry is 21, and with increasing crystal symmetry the redundancies become more numerous; only three independent compliances are needed to describe the elastic properties of a cubic crystal.

Stress

Stress is a tensor quantity that describes the mechanical force density (force per unit area) on the complete surface of a domain inside a material body. A stress exists wherever one part of a body exerts a force on neighboring parts. Its orientation is not tied to any particular directions that are intrinsic to the material like, say, crystallographic axes. It is thus distinct from the matter tensors that were discussed in the preceding chapter, all of which have definitive orientations within a crystal or other anisotropic material; it is called a field tensor (and so is strain). The definition of stress depends on the concept of a continuum. Let f(xi) be a single-valued function defined for every point xi in a region. This function is said to be continuous at the point xi if the following holds for all paths of approach of xi to °xi:. . . f(xi) → f(°xi) as xi → °xi (4.1)· . . . Equivalently, for any number ∊, no matter how small, there exists a neighborhood of nonzero radius around the point xi in which: . . . . 〈f(xi) − f(°xi)〉2 < ∊,­ (4.2) . . . for all points xi in that neighborhood. A continuum is an idealized material whose physical attributes are continuous functions of position. Thus neighboring points remain neighbors, and a continuum cannot have gaps or jumps (discontinuities) in its properties. Surfaces bounding gaps or defining discontinuities must be specially treated in continuum mechanics. Examples are surfaces between two fluids of differing density or viscosity, or between solids with different thermal conductivity or elastic properties. Real materials are never continua; they are discontinuous at the atomic scale, and often at larger scales as well. The notion of a continuum is, therefore, only a macroscopic approximation, but it allows useful mathematical approaches to the treatment of real phenomena.

Fields

A scalar determined at every point in a given domain, analytically or otherwise, constitutes a scalar field. Vectors similarly determined constitute a vector field. The defining analytical expressions of a three-dimensional field are commonly differentiable with respect to space; hence in a cartesian coordinate system they are amenable to partial differentiation with respect to x1, x2, and x3. In this context it is useful to define several differential operators. The operator ∇ is called the “del” or the “nabla” and is defined as follows: . . . ∇ ≡ i 𝜕/𝜕x1+j 𝜕/𝜕x2+k 𝜕/𝜕x3, (2.1) . . . or: . . . (∇)i ≡ 𝜕/𝜕xi. (2.2) . . . It can be seen that the del is a vector. By convention, however, it is not rendered in boldface. Before we define additional differential operators, we extend the subscript notation further and let a subscribed comma indicate partial differentiation. A comma preceding a letter subscript, say i, is taken to imply differentiation with respect to xi. Thus, if φ (xi) is a scalar function of position and thus defines a scalar field, its gradient, another differential operator, is defined by the equation: . . . (grad φ)i ≡ φ,i≡ 𝜕 φ/𝜕xi. (2.3) . . . Thus the gradient of a scalar is a vector.

Finite Strain

Before dealing with finite strain, we demonstrate an important property of the determinant of a three by three matrix the rows of which are vectors. The result can be used to calculate the volume of a general parallelepiped (a solid bounded by parallelograms). The following problems also introduce the concept of dual position vectors, an original set, ai, and a final set, xi, where one set is a function of the other. This concept may alternatively be taken to produce two distinct cartesian coordinate systems, a concept used throughout this chapter. The description of a transformation from coordinates ai to xi, is designated as Lagrangian if it is referred to the system of original coordinates ai and as Eulerian if it is referred to the system of final coordinates xi. The origins and orientations of the two coordinate systems are independent of each other. For certain purposes, however, it is useful to let the two coordinate systems coincide in origin and orientation, being distinct only by the fact that they indicate positions before and after some event (such as a deformation).

Matter Tensors and Coordinate Transformations

Vectors, the subject of the previous two chapters, may be classified as members of a class of mathematical entities called tensors, insofar as they can be expressed in the form of ordered arrays, or matrices, and insofar as they further conform to conditions to be explored in the present chapter. Tensors can have various ranks, and vectors are tensors of the first rank, which in three-dimensional space have 31 or three components. Much of this, and later, chapters deals with tensors of the second rank which in the same space have 32 or nine components. Tensors of higher (nth) rank do exist and have 3n components, and so do, at least nominally, tensors of zero rank with a single, or 30, component, which makes them scalars. Tensors of the second rank for three dimensions are written as three-by-three matrices with each component marked by two subscripts, which may be either letters or numbers.