AME/BME 466/566 - 2013 - Biomechanical Engineering Lecture 4: Stress and tensors, definition of Cauchy stress

Stress, strain and constitutive equations • we wish to describe the relationship between motion or deformation of materials and forces acting on them • consider first a simple spring: an applied force F causes an extension d • by experiment, F = K × d (Hooke's law) where K is the spring constant

• three key elements in this equation - F, a measure of force - d, a measure of displacement - K, a measure of material properties • now consider a continuum: deformations and forces depend on position • stress is a measure of forces acting in a continuum • strain is a measure of displacements and deformations of a continuum • rate of strain is a measure of rate of deformation of a continuum • constitutive equation states how the stress depends on the strain and/or rate of strain, and depends on the physical properties of the material

Stress • a measure of forces acting in a continuum • for a particle, we can describe the forces by a vector • in a continuum, each part exerts forces on neighboring parts

• consider a small surface in the fluid with area ∆ S and normal direction n • suppose F is the force exerted by the material above on the material below • define stress vector or traction T = force per unit area = F/∆ S • units are dyn/cm2 or Pa = newton/m2 • can resolve into normal and shear components

1 Components of stress • the stress vector T depends on the orientation of the surface • the stress vector on any surface is completely determined if the stress vectors on three perpendicular surfaces are known • consider Cartesian coordinates with vectors x1, x2, x3 along the axes • suppose that the three surfaces are normal to x1, x2, x3 • then the nine components of the corresponding stress vectors are:

• τ11, τ22, τ33 are normal components and τ12, τ13, τ23, etc. are shear components

Calculation of the stress acting on a surface with any other orientation • suppose that the unit normal to the surface is given by n = (n1, n2, n3) • then the stress vector is given by a linear combination: T = n1 (τ11, τ12, τ13) + n2 (τ21, τ22, τ23) + n3 (τ31, τ32, τ33) = (T1, T2, T3) where T1 = n1 τ11 + n2 τ21 + n3 τ31, T2 = n1 τ12 + n2 τ22 + n3 τ32, T3 = n1 τ13 + n2 τ23 + n3 τ33 • writing out all the components is tedious, so we use index notation Tj = n1 τ1j + n2 τ2j + n3 τ3j for j = 1, 2, 3 • and the summation convention - any index that appears twice in a single quantity or a product of quantities implies a summation over that index - so Tj = ni τij

Example of index notation and summation convention • if a = (4,5,6) and b = (1,0,-1) then a1 = 6, a2 = 5, a3 = 6 and b1 = 1, b2 = 0, b3 = −1 • ai bi = a1 b1 + a2 b2 + a3 b3 = 4×1 + 5×1 + 6×(−1) = −2

2 Properties of the stress tensor • the matrix of components τij forms a tensor - a generalization of a vector • known as the Cauchy stress tensor or simply as the stress tensor • other notations are σij and Tij • the tensor is second rank: it has two subscripts, i.e., each component has two directions associated with it (normal and stress vector) • it must satisfy equilibrium equations, obtained by considering the forces acting on a small control volume δx1 × δx2 × δx3 (Cauchy’s first law)

∂τ11 ∂τ12 ∂τ13 + + + X1 = 0 ∂x1 ∂x 2 ∂x 3 ∂τ ∂τ21 ∂τ22 ∂τ23 ij + + + X 2 = 0 i.e., + X i = 0 ∂x1 ∂x 2 ∂x 3 ∂x j

∂τ31 ∂τ32 ∂τ33 + + + X 3 = 0 ∂x1 ∂x 2 ∂x 3 using the summation convection, where Xi are the components of the body force acting on the material (e.g. gravity). Here, inertial (acceleration) terms are neglected. • it is symmetric under most conditions: τ21 = τ12, τ23 = τ32, τ13 = τ31 (Cauchy’s second law) • it satisfies the transformation law for change of axes: the components in another set of axes (x'1, x'2, x'3) are given by τ'km = τji βkj βmi where βik is the direction cosine of the x'k axis with respect to the xi axis, i.e., the cosine of the angle between them. • other forms of the stress tensor are the first and second Piola-Kirchhoff stress tensors, to be discussed later

Notations for tensors • a second-rank tensor can be represented as: • a bold symbol - T • or a subscripted symbol - Tij where i and j can take the values 1, 2 or 3  T T T   11 12 13  • or an array (matrix) of values T21 T22 T23    T31 T32 T33  • the subscripted symbol (using index notation and the summation convention) is compact but avoids ambiguity when computing with tensors

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