Appendix Fundamentals of Strength of Materials In this chapter some basics of higher Theory of Strength of Materials are presented. It is a compressed compilation of the essential concepts and relationships that are necessary for understanding the book without giving derivations. For the reader interested in detailed studies, advanced textbooks on strength of materials, continuum mechanics and material modeling are recommended. A.1 Mathematical Representation and Notation The mathematical representation of vectors and tensors as well as algebraic and analytic calculation rules are given both in symbolic form (bold italics) and in index notation. We follow the generally accepted rules and notation such as Einstein’s sum- mation convention and comma form of differentiation. The index notation is basically restricted to a Cartesian, space-fixed coordinate system. Upper-case letters for vari- ables and indices generally refer to the reference configuration, whereas lower-case letters mark the current configuration. Matrices are written bold but upright. → 3 Vector: a =a = a1e1 + a2e2 + a3e3 = ai ei = ai ei i=1 ⇒ 3 3 Tensor second order: A = A = Aijei e j = Aijei e j i=1 j=1 Tensor 4th order: C=Cijklei e j ek el A, A, A Invariants of a tensor A: I1 I2 I3 Unit tensor second order: I = δijei e j with Kronecker-symbol: δij Scalar product: a · b = ai bi Double scalar product: A : B = AijBij T Column matrix (m × 1): a = [a1a2 ···am] M. Kuna, Finite Elements in Fracture Mechanics, Solid Mechanics and Its 391 Applications 201, DOI: 10.1007/978-94-007-6680-8, © Springer Science+Business Media Dordrecht 2013 392 Appendix: Fundamentals of Strength of Materials ⎡ ⎤ A11 A12 ···A1n ⎢ ⎥ ⎢ A21 A22 ···A2n ⎥ Matrix (m × n): A = ⎢ . ⎥ ⎣ . ⎦ Am1 Am 2 ··· Amn Transpose/inverse matrix: AT , A−1 , A−1 T = A−T Variational symbol: δ Partial derivative: ∂(·)/∂xi = (·),i ∂(·) ∂(·) ∂(·) Nabla-operator: ∇(·) = e1 + e2 + e3 = (·),i ei ∂x1 ∂x2 ∂x3 ∂2(·) ∂2(·) ∂2(·) Laplace-operator: (·) = + + = (·), ∂ 2 ∂ 2 ∂ 2 ii x1 x2 x3 A.2 State of Deformation A.2.1 Kinematics of Deformation As a result of loading, every deformable body experiences a motion in space and time as illustrated in Fig. A.1. The kinematics of continua deals with the geometrical aspects of the motion. The entire motion is composed of translation and rotation of the body as a whole (rigid body motions) as well as the relative displacement of its particles (deformation). For this purpose it is necessary to assign at any time to each particle (material point) of the body a corresponding location in physical space (spatial point). Such a one-to-one mapping defines a configuration of the body. The undeformed state of the body at time t = 0 is denoted as reference configuration comprising the volume V and the surface A. The location of each particle P is Reference configuration t = 0 Current configuration t > 0 A V v Q uu+ d d X P Q’ a dx X u P’ e3 xX(), t e2 e1 Fig. A.1 Kinematics of deformable body A.2 State of Deformation 393 identified by its position vector X = X M eM in a Cartesian coordinate system with basis vectors eM . These material coordinates do not change during the motion, but mark uniquely each particle and are also called Lagrangian coordinates. At a later time t > 0 of motion, the deformed body occupies the current configuration with the changed volume v and surface a. The particle P runs along the trajectory to P, its current position vector x = xm em is described by the spatial coordinates xm (Eulerian coordinates). The motion of a body is thus the temporal sequence of such configurations. The function x = x(X, t) (A.1) describes the position of the particles P in space, whereby the time t is the curve parameter and the initial position X is the group parameter. During the motion each material point P is shifted by the vector u to the position P.Thisdisplacement vector marks the difference between the current position x of a particle and its initial one X. u(X, t) = x − X or um(X, t) = xm − Xm x(X, t) = X + u or xm(X, t) = Xm + um (A.2) Hence, the deformation state of a body is uniquely characterized in space and time by the displacement field u(X, t). The physical properties of the particles (density, temperature, material state, etc.) and the field quantities yet to be determined by the initial boundary value problem (displacements, strains, stresses, etc.) change during the motion. To describe the temporal changes of these field quantities χ, there are two fundamentally different approaches. In the Lagrangian (material) description the changes of χ are traced for each particle and are characterized by the functional dependence on the material coordinates X. χ = χ(X, t) (A.3) The observer is quasi attached to the moving particle and measures the changes of the field variables. In the Eulerian (spatial) description the changes of a field quantity χ are observed at a fixed spatial point and are therefore expressed as a function Ξ of the spatial coordinates x. χ = Ξ(x, t) (A.4) An observer fixed at the position x measures the changes resulting from the fact that unlike particles with different properties are passing by. In principle both formula- tions are equivalent and can be converted into each other, if the motion is known. The Eulerian description is preferred in fluid mechanics, because there changes of field quantities (pressure, velocity) at fixed positions are commonly of interest. The Lagrangian description has advantages in solid mechanics, because here the initial state is usually known and the individual properties of the particles (deformation, stress or state variables) need to be traced during the loading history. 394 Appendix: Fundamentals of Strength of Materials A.2.2 Deformation Gradient and Strain Tensors To characterize the stress in the interior of a solid body, the relative displacements between two infinitesimally neighboring particles P and Q are of primary impor- tance. In Fig. A.1 a material line element PQ = dX = dX M eM is considered in the initial configuration, which is deformed into P Q = dx = dxm em during its motion in the current configuration. ∂xm(X, t) dxm = dX M = FmM dX M or dx = F · dX (A.5) ∂ X M The partial derivatives of xm with respect to X M form a tensor of second order that is called a deformation gradient F. It can be expressed by the displacement field and the unit tensor of second order I using (A.2). ∂xm ∂um FmM = = δmM + , I = δmMem eM (A.6) ∂ X M ∂ X M The mathematical inversion of (A.5) represents the deformation of the reference configuration as viewed from the current configuration. ∂ ( , ) = X M x t = −1 = −1 · dX M dxm F Mm dxm or dX F dx (A.7) ∂xm ∂ ∂u −1 = X M = δ − M FMm Mm (A.8) ∂xm ∂xm The deformation gradient provides the relationship between the material line ele- ments in the initial and the current configuration. It is defined on the basis vectors of both configurations (two-point tensor) and in general not symmetric. To make the affine mapping X ↔ x unique and reversible, the Jacobian functional determinant J must be definite. ∂xm J = det = det[FmM] = 0(A.9) ∂ X M The deformation gradient involves both the elongation and the local rigid body rota- tion in the vicinity of a material point X.ByPQ = dX an arbitrary direction in the neighborhood of P is defined. The elongation and the local rotation of the line element can be separated from each other by means of the polar decomposition: F = R · U = V · R or FkN = RkM UMN = Vkm RmN. (A.10) The rotation is described by the orthogonal tensor R, T = −1 = −1 = , R R or RLk RkL and det [RkL] 1 (A.11) A.2 State of Deformation 395 whereas both stretch tensors U and V quantify solely the stretch (final length / initial length) of the line elements. The mapping F can be imagined as a sequence of a rotation R and an elongation V or as an elongation U followed by a rotation R.The tensors U and V are symmetric and positive definite. U = UMN eM eN right stretch tensor (reference configuration) V = Vmn em en left stretch tensor (current configuration) V = R · U · RT and U = RT · V · R (A.12) To split off the less interesting rotation term, the so-called right and left Cauchy- Green deformation tensors are introduced, which are obtained from the stretch tensors or straight from the deformation gradient F as follows: T T C = F · F = U · U or CMN = FkM FkN = ULM ULN (right) T T b = F · F = V · V or bmn = FmL FnL = Vml Vnl (left) (A.13) All measures of deformation U, V, b and C migrate into the unit tensor I if no stretching occurs. The squares of the arc lengths of material line elements are calculated in the reference configuration as − − (dL)2 = dX · dX = (F 1 · dx) · (F 1 · dx) = · ( −T · −1) · = · −1 · = −1 dx F F dx dx b dx dxk bkl dxl (A.14) and in the current configuration as (dl)2 = dx · dx = (F · dX) · (F · dX) T = dX · (F · F) · dX = dX · C · dX = dX K CKL dX L . (A.15) The elongation of the line element amounts to (dl)2 − (dL)2 = 2dX · E · dX = 2dx · η · dx . (A.16) This way the following strain measures are defined: Green-Lagrangian strain tensor (related to the reference configuration): 1 1 E = (C − I) or EKL = (CKL − δKL) (A.17) 2 2 396 Appendix: Fundamentals of Strength of Materials Euler-Almansi strain tensor (related to the current configuration): 1 −1 1 −1 η = (I − b ) or ηkl = (δkl − b ) (A.18) 2 2 kl Both strain tensors yield the information about the relative changes of lengths and angles of material line elements in the vicinity of a point P due to deformation.