Cauchy stress defined

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

The spatial Cauchy stress tensor s is defined at time by f = sn, where t(x, t, n) is a contact force vector acting on an element of area da = n da with unit normal i and magnitude da in the current configuration. The element of area... 

The volumetric constitutive equations for a chemoporoelastic material can be formulated in terms of the stress S = a,p, it and the strain 8 = e, (, 9, i.e., in terms of the mean Cauchy stress a, pore pressure p, osmotic pressure it, volumetric strain e, variation of fluid content (, and relative increment of salt content 9. Note that the stress and strain are measured from a reference initial state where all the stress fields are equilibrated. The osmotic pressure it is related to the change in the solute molar fraction x according to 7r = N Ax where N = RT/v is a parameter with dimension of a stress, which is typically of 0( 102) MPa (with R = 8.31 J/K mol denoting the gas constant, T the absolute temperature, and v the molar volume of the fluid). The solute molar fraction x is defined as ms/m with m = ms + mw and ms (mw) denoting the moles of solute (solvent) per unit volume of the porous solid. The quantities ( and 9 are defined in terms of the increment Ams and Amw according to... 

The first two terms on the right-hand side of equation [12.6] are viscoelastic terms proposed by Schapery, where e represents uniaxial kinematic (or total) strain at time t, o is the Cauchy stress at time t, is the instantaneous compliance and AD(r[i ) is a transient creep compliance function. The factor g defines stress and temperature effects on the instantaneous elastic compliance and is a measure of state dependent reduction (or increase) in stiffness. Transient compliance factor gi has a similar meaning, operating on the creep compliance component. The factor gj accounts for the influence of loading rate on creep. The function i ) represents a reduced timescale parameter defined by ... 

Because of chain inextensibility, the shear rate of any slip system is not dependent on the normal-stress component in the chain direction (Parks and Ahzi 1990). This renders the crystalline lamellae rigid in the chain direction. To cope with this problem operationally, and to prevent global locking-up of deformation, a special modification is introduced to truncate the stress tensor in the chain direction c. Thus, we denote by S° this modification of the deviatoric Cauchy stress tensor S in the crystalline lamella to have a zero normal component in the chain direction, i.e., by requiring that 5 c,c = 0, where c,- and c,- are components of the c vector (Lee et al. 1993a). The resolved shear stress in the slip system a can then be expressed as r = where R is the symmetrical traceless Schmid tensor of stress resolution associated with the slip system a. The components of the symmetrical part of the Schmid tensor / , can be defined as = Ksfw" + fs ), where if and nj are the unit-vector components of the slip direction and the slip-plane normal of the given slip system a, respectively. 

In strength of materials text, it is weU known that Cauchy stress, small strain because the area in the undeformed body and deformed body is almost the same. For a large strain, however, we generally do not know the area of the deformed configuration. Thus we need to define a stress measure that we can use in the reference configuration. However, it is noted that Cauchy stress is still the most used stress definition or tme stress because the equilibrium is about the deformed body but not the undeformed body. Therefore, other definitions of stress are for convenience of mathematical operation. Stress is generally reported as the Cauchy stress. This shows a distinct departure from the strain definitions. 

The Cauchy stress tensor , a, at any point of a body (assumed to behave as a continuum) is completely defined by nine component stresses-three orthogonal, normal stresses and six orthogonal, shear stresses. It is used for the stress analysis of materials undergoing small deformations, in which the differences in stress distribution, in most cases, can be neglected. 

Newton s second law states that in an inertial frame the rate of linear momentum is equal to the applied force. Here, by applying the second law to a continuum region, we define the Cauchy stress, and derive the equation of motion. 

We introduce a relationship between the Cauchy stress a defined in the deformed body with its basis e, and the first Piola-Kirchhoff stress II defined in the undeformed body with its basis Ej as follows ... 

During the motion of the body, its volume, surface area, density, stresses, and strains change continuously. The stress measure that we shall use is the 2nd Piola-Kirchhofif stress tensor. The components of the 2nd Piola-Kirchhoff stress tensor in Cj will be denoted by To see the meaning of the 2nd Piola-KirchhofiF stress tensor, consider the force dF on surface dS in C2. The Cauchy stress tensor t is defined by... 

The stress tensor has been introduced in Chapter 2. In small strain elasticity theory, the components of stress are defined by considering the equilibrium of an elemental cube within the body. When the strains are small, the dimensions of the body, and therefore the areas of the cube faces, are to a first approximation unaffected by the strain. It is then of no consequence whether the components of stress are defined with respect to the cube before deformation or the cube after deformation. For finite strains, however, this is not true and there are alternative definitions of stress depending on whether the deformed or undeformed state is chosen as a reference. We will choose to adopt the stress associated with the deformed state - the true stress or Cauchy stress - throughout this work. In our present axis notation, we can express this stress tensor X as... 

P.3. 1) There exists a stress constitutive functional T O S which gives the current value of Cauchy stress T corresponding to the history F and it is defined by... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

The stress and strain tensors aij u),Sij u) are defined by the Hooke and Cauchy laws... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

The average force per unit area is Af,-/AS. This quantity attains a limiting nonzero value as AS approaches zero at point P (Cauchy s stress principle). This limiting quantity is called the stress vector, or traction vector T. But T depends on the orientation of the area element, that is, the direction of the surface defined by normal n. Thus it would appear that there are an infinite number of unrelated ways of expressing the state of stress at point P. 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

The stress vector represents the force of the material outside of V(t) acting on the material inside V(t). The assumption, based on the short range of the inter-particle forces, is that this force which the material exerts on itself acts entirely on the surface S(t). Truesdell calls this stress principle the defining concept of continuum mechanics. The stress vector depends not only on time and space, but also on the surface orientation. However, according to Cauchy s fundamental theorem,... 

The simplest (and weakest ) definition of an elastic material is one for which the stress depends only on the current strain these materials are termed Cauchy elastic. A subset of these materials is occupied by those for which the strain energy depends only on the current strain. These are termed Green elastic or hyperelastic and for these the strain energy is a function of the current strain only, and fully defines the material behaviour. For Cauchy... 

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. 

Another special case of the Rivlin-Sawyer model that is a generalization of Eq. 10.10 was proposed by Wagner and Demarmels [12] who added a dependency on the Cauchy tensor to Eq. 10.10 to provide for a non-zero second normal stress difference and a better fit to data for planar extension, which is defined in Section 10.9. Their model is shown as Eq. 10.11. 

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