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# Stress strain relationships in rocks

This page provides an introduction to stress-strain relationships. They form the foundation for several rock properties such as elastic moduli (incompressibility), effective media theory, elastic wave velocity, and rock strength.

## Defining stress and pressure

Stress is the force per unit area. ....................(1)

The metric units of stress or pressure are N/m2 or Pascals (Pa). Other units that are commonly used are bars, megapascals (MPa), and lbm/in.2 (psi) (see Table 1).

Stresses can take various forms such as:

• Homogeneous pressure P
• Normal stress σn
• Stress applied at a general angle σg

These are illustrated in Fig. 1.

General stress can be decomposed into normal and tangential components (Fig. 1c). We usually refer to balanced stresses because, under quasi-static conditions, they produce no net acceleration. Stress is a second-order tensor denoted by σij, where the first index denotes the surface and the second the direction of the applied force (see Fig. 2). In earth sciences and engineering, compressive stresses are usually considered positive, whereas most material sciences consider tensional stress positive. More details on the influence of stresses and the stress tensor can be found in Jeager and Cook and Nye.

### Standard stress conditions

Several standard stress conditions are either assumed for analysis or modeling, or applied in the laboratory:

• Hydrostatic stress - all confining stresses are equal
• Uniaxial stress - one stress applied along a single axis (other stresses are zero or held constant during an experiment)
• Biaxial stress - two nonequal stresses applied (third direction is equal to one of the others)
• Triaxial stress - while this really represents three independent principal stresses, it is commonly used to represent separate vertical and two equal horizontal stresses (e.g., biaxial)
• Anisotropic stresses are usually responsible for rock deformation and failure (see Rock failure relationships).

We will concern ourselves primarily with mean stress (σm) or pressure (P). ....................(2)

### Types of pressure

It is important to distinguish among the various kinds of pressure, because the combination often determines any specific rock property and influences the response to any production procedure.

 Confining pressure = Pc = Overburden  pressure  on  rock  frame Pore pressure = Pp = Fluid  pressure  inside  pore  space Differential (or net) pressure = Pd = Difference  between  Pc and Pp Effective pressure = Pe = Combination  of  Pc  and  Pp  controlling  a  property

Increasing confining pressure (Pc) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (Pp) tends to increase rock volume. Pp counteracts the effects of Pc. Thus, rock properties are controlled largely by the difference between Pc and Pp, or the differential pressure Pd. A more exact form will account for the interaction of the fluid pressure with the pore space and minerals and result in an effective stress (Pe) law ....................(3)

where n is a term that can be derived theoretically or defined experimentally for each property.

## Deformation, strain and modulus

Application of a single (vertical) stress is one typical experiment run to measure material mechanical properties (Fig. 3). If this stress continues to increase, eventually the material will fail when the uniaxial compressive strength is reached (see Rock failure relationships). For the rest of this topic, however, we will deal only with small deformations and stresses such that the rock remains in the elastic region.

### Vertical strain

Restricting ourselves to small deformations and stresses, several important material properties can be defined. For an isotropic, homogeneous material, there is a vertical deformation (ΔL) associated with the vertical stress. Normalizing this deformation by the original length of the sample, L, gives the vertical strain ....................(4)

### Young's modulus

By definition, Young’s modulus, E, is the ratio of the applied stress (σzz) to this strain ....................(5)

Because strain is dimensionless, E is in units of stress.

### Lateral strain

This same stress will generally result in a lateral or horizontal deformation, ΔW. The lateral strain can then be defined ....................(6)

### Poisson's ratio

One important parameter relating the vertical and horizontal strains is Poisson’s ratio ....................(7)

The minus sign is attached because the signs of the deformations are opposite for the horizontal vs. vertical strains in this simple case.

### Volumetric strain

If instead we applied a pressure, we would get a volumetric strain εv: ....................(8)

### Bulk modulus

The bulk modulus of a material is then defined as the ratio of applied pressure to volumetric strain ....................(9)

Bulk modulus is equivalent to the inverse of compressibility, β.

### Shear modulus

In a similar way, shear modulus, μ (often "G" in many publications), can be defined as the ratio of shear stress to shear strain: ....................(10)

### Hooke's law

These various equations are special cases of Hooke’s Law, which can be written for the general case ....................(11)

Stress and strain are both tensors with 9 components. Cijkl would then be a tensor with 81 components. However, because of symmetry considerations, only a maximum of 21 can be independent (a thorough treatment of the tensor relations is provided in Nye). For isotropic materials, this reduces to ....................(12)

where λ is Lame’s constant. In fact, for isotropic materials, there are only two independent elastic parameters. Any isotropic elastic constant can be written in terms of two others. For example, λ can be defined as ....................(13)

The possible combinations among various isotropic elastic constants are shown in Table 2. This becomes important in applications, because restricting one term, say ν, fixes the ratio of other moduli such as μ and K.

## Nomenclature

 λ = Lame’s parameter, GPa or MPa, Eq. 13 μ = shear modulus, GPa or MPa, Eq. 10 Cijkl = stiffness tensor components, GPa or MPa E = Young’s modulus, GPa or MPa K = bulk modulus, GPa or MPa L = length, m ΔL = change in length, m n = number of moles P = pressure, MPa Pc = confining pressure, MPa Pd = differential pressure, MPa Pe = effective pressure, MPa Pp = pore pressure, MPa Vrock = rock velocity, m/s εij = strain components, fractional εkl = strain components, fractional εshear = shear strain, fractional εV = volumeteric strain, fractional εyy = horizontal strain, fractional εzz = vertical strain, fractional σij = stress components, GPa or MPa σm = mean stress, GPa or MPa σn = normal stress, GPa or MPa σshear = shear stress components, GPa or MPa σzz = vertical stress component, GPa or MPa δ = loss tangent