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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.
Contents
Defining stress and pressure
Stress is the force per unit area.
The metric units of stress or pressure are N/m^{2} 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 σ_{n}
- Stress applied at a general angle σ_{g}
These are illustrated in Fig. 1^{[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 σ_{ij}, 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^{[2]} and Nye.^{[3]}
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 (σ_{m}) or pressure (P).
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 | = P_{c} | = Overburden pressure on rock frame |
Pore pressure | = P_{p} | = Fluid pressure inside pore space |
Differential (or net) pressure | = P_{d} | = Difference between P_{c} and P_{p} |
Effective pressure | = P_{e} | = Combination of P_{c} and P_{p} controlling a property |
Increasing confining pressure (P_{c}) alone will result in a decrease of rock volume, or compaction. In contrast, increasing the pore pressure (P_{p}) tends to increase rock volume. P_{p} counteracts the effects of P_{c}. Thus, rock properties are controlled largely by the difference between P_{c} and P_{p}, or the differential pressure P_{d}. 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 (P_{e}) law
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
Young's modulus
By definition, Young’s modulus, E, is the ratio of the applied stress (σ_{zz}) to this strain
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
Poisson's ratio
One important parameter relating the vertical and horizontal strains is Poisson’s ratio
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 ε_{v}:
Bulk modulus
The bulk modulus of a material is then defined as the ratio of applied pressure to volumetric strain
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:
Hooke's law
These various equations are special cases of Hooke’s Law, which can be written for the general case
Stress and strain are both tensors with 9 components. C_{ijkl} 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^{[3]}). For isotropic materials, this reduces to
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
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 |
C_{ijkl} | = | 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 |
P_{c} | = | confining pressure, MPa |
P_{d} | = | differential pressure, MPa |
P_{e} | = | effective pressure, MPa |
P_{p} | = | pore pressure, MPa |
V_{rock} | = | rock velocity, m/s |
ε_{ij} | = | strain components, fractional |
ε_{kl} | = | strain components, fractional |
ε_{shear} | = | shear strain, fractional |
ε_{V} | = | volumeteric strain, fractional |
ε_{yy} | = | horizontal strain, fractional |
ε_{zz} | = | vertical strain, fractional |
σ_{ij} | = | stress components, GPa or MPa |
σ_{m} | = | mean stress, GPa or MPa |
σ_{n} | = | normal stress, GPa or MPa |
σ_{shear} | = | shear stress components, GPa or MPa |
σ_{zz} | = | vertical stress component, GPa or MPa |
δ | = | loss tangent |
References
- ↑ ^{1.0} ^{1.1} Hubbert, M.K. 1937. Theory of scale models as applied to the study of geologic structures, No. 48. New York: Bulletin of the Geological Society of America, Geological Society of America.
- ↑ Jaeger, J.C. and Cook, N.G.W. 1979. Fundamentals of Rock Mechanics, third edition. London: Chapman and Hall.
- ↑ ^{3.0} ^{3.1} Nye, J.F. 1972. Physical Properties of Crystals. London: Oxford University Press.
Noteworthy papers in OnePetro
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
External links
Aldrich, M.J. 1969. Pore Pressure Effects on Berea Sandstone Subjected to Experimental Deformation. Geol. Soc. Am. Bull. 80 (8): 1577-1586. http://dx.doi.org/10.1130/0016-7606(1969)80[1577:ppeobs]2.0.co;2
See also
Rock moduli boundary constraints
Isotropic elastic properties of minerals
Pore fluid effects on rock mechanics
Rock strength from log parameters