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Theory of Elasticity
4
CHAPTER
Theory of Elasticity 4.1 INTRODUCTION Many engineering components, while in service, are subjected to a variety of loading. Examples include an airplane wing, which is subjected to lift and drag forces and formation rocks, which are subjected to pore pressure, in-situ stresses and the forces imposed by a drilling bit. In such situations it is necessary to know the characteristics of the engineering component’s materials such that any resulting deformation will not be excessive and failure will not occur. This is achieved by means of stress-strain relationships. The theory of elasticity a methodology that creates a linear relation between the imposing force (stress) and resulting deformation (strain), for the majority of materials which behave fully or partially elastically. It plays an important role in the design of man-made components and structures as well as the assessment of the integrity of stable natural systems that have been disturbed by man. The principles of the theory of elasticity and relevant stress-strain equations are discussed in this chapter, as prerequisites to studying failure criteria in Chapter 5.
4.2 MATERIALS BEHAVIOR The degree to which a structure strains depends on the magnitude of an imposed load or stress. The element of stress is therefore very central in solids mechanics. However, stress cannot be measured directly. Usually strain (deformation) is measured in-situ or in the laboratory, and the stresses are then calculated. For many materials that are stressed in tension or compression at very low levels, stress and strain are proportional to each other through a simple linear relationship as shown in part (a) of Figure 4.1. However, the stress-strain (force-deformation) relation is not always simple and linear, and it can change with change of material properties and
Petroleum Rock Mechanics DOI 10.1016/B978-0-12-385546-6.00004-8
2011 Elsevier Ltd. All rights reserved.
|
41
42
Materials Behavior
[(Figure_1)TD$IG]
Figure 4.1 Typical stress-strain behavior of a material from onset of loading to fracture: (a) elastic deformation, (b) early plastic deformation, (c) further plastic deformation, (d) failure due to plastic deformation. (Callister, 2000).
geometry. Stress-strain relations, known also as constitutive relations, are normally found empirically. Consider a rod with initial length lo in tension, as shown in Figure 4.2a. Having applied an axial load to the rod, it stretches to an additional length Dl = l lo. It can be seen that the rod elongates in the axial direction but to minimize volume increase, it will become thinner in the middle section.
[(Figure_2)TD$IG]
Figure 4.2 Stress-strain diagram showing linear elastic deformation.
Theory of Elasticity
43
4.3 HOOKE’S LAW A linear relationship exists between the stress and strain as shown in Figure 4.2b and is given by the following equation: sx ¼ E«x
ð4:1Þ
where: sx ¼
Fx A
and the engineering strain definition of «x, i.e. Equation 1.4, is used: «x ¼
Dl lo
Equation 4.1 is known as Hooke’s law of deformation, and the slope of the stress-strain diagram is referred to as the Modulus of Elasticity (Young’s modulus), E. Using sx and «x in Equation 4.1, Hooke’s law can also be expressed as: Dl ¼
F x lo EA
ð4:2Þ
The tensor representation of Hooke’s law is given by: sij ¼ L ijkl «kl
ð4:3Þ
where all stress and strain components are coupled with anisotropic properties. It should be noted that the simplest possible representation should always be used, as the above equation can be very complex and difficult to solve. The ratio of traverse strain «y to axial strain «x is defined as the Poisson’s ratio, and is expressed by: n¼
«y «x
ð4:4Þ
4.4 HOOKE’S LAW IN SHEAR The properties of materials in shear can be evaluated from shear tests or by other methods such as a torsion test. In either method, a plot of shear stress
44
Analysis of Structures
versus shear strain is produced, which is similar to the strain-stress plot that results from a normal tension test (see Figure 4.1). This is known as shear stress-shear strain relation. From the stress-strain plot, we can determine properties such as the modulus of elasticity, yield strength and ultimate strength. These properties in shear are usually about half the size of similar properties in tension, and they are evaluated using a shear stress-shear strain plot. For many materials, the initial part of a shear stress-shear strain plot is a straight line, which is the same as the plot that results from a tension test. For this linearly elastic region, the shear stress and shear strain are proportional, and can therefore be expressed in a similar form to Equation 4.1 as given below: t ¼ Gg
ð4:5Þ
where t is the shear stress, G is shear modulus or modulus of rigidity and g is the shear strain. Equation 4.5 is the one dimensional Hooke’s law in shear.
4.5 ANALYSIS OF STRUCTURES Structures can be categorized into two main groups: (i) statically determinate structures and (ii) statically indeterminate structures. The first group’s feature is that their reactions and internal forces can be determined solely from free-body diagrams and the equation of equilibrium. The second group of structures is, however, more complex, and their reactions and internal forces cannot be evaluated by the equation of equilibrium alone. To analyze such structures we must provide additional equations pertaining to its deformation/displacement. The following equations must be satisfied by a statically indeterminate structure and have to be solved simultaneously: Equation of Equilibrium: This equation results from a free body diagram where a relationship exists between applied forces, reactions and internal forces. Equation of Compatibility: This equation expresses the fact that the changes in dimensions must be compatible with the boundary conditions. Constitutive Relation (Stress-Strain Equation): This relation expresses the link between acting forces (stresses) and deformations/displacements (strains). This relation, as explained earlier, has various forms depending upon the properties of material.
Theory of Elasticity
45
4.6 THEORY OF INELASTICITY Referring to the definition of an elastic body as one in which the strain at any point on the body is linearly related to and completely determined by stress, then an obvious definition of an inelastic body is one in which the stress-strain relationship is not linear due to additional elements which may affect the material’s behavior. Theory of inelasticity is complex, and very much dependent on the behavior of the material. However, for materials that have two rather distinct elastic and inelastic regions, separated by a yield strength point, the theory of inelasticity can be simplified by using a continuous function to approximate the stress-strain relation over both the elastic and inelastic regions. In this model, the stress strain relation is approximated by two straight lines, one describing slope E as defined by Hooke’s law in Equation 4.1, and the other with slope aE as given below: sx ¼ ð1 aÞS y þ aE«x
ð4:6Þ
where Sy is the yield strength, E is the modulus of elasticity and a is the strain hardening factor. The linear elastic and inelastic relation offers many advantages, especially because it results in an explicit mathematical formulation and can therefore be used for most applications. More complex, material formulations are introduced when certain applications are required. These relations are dependent strongly upon the deformation of the material.
4.7 CONSTITUTIVE RELATION FOR ROCKS In Section 4.2, a linear stress-strain model was shown as being a usual case for metals. It should however be noted that at high loads, metals may yield before failure, and for some applications a more accurate relation may be required. Therefore, in Section 4.6 an additional, simplified, linear stress-strain model was developed to account for the inelastic behavior of a metal. Rocks behave similarly to brittle metals; i.e. they deform linearly at small loads, but in a non-linear or plastic fashion at higher loads. Therefore, the combined, linearized elastic and inelastic model may represent many materials, including rocks, with a reasonable accuracy. However, depending on the rock materials, sometimes a more accurate
46
Constitutive Relation for Rocks
[(Figure_3)TD$IG]
Figure 4.3 Schematic plots of different constitutive relationships; (a) perfect elastic material, (b) real material in tension, (c) elastoplastic material, (d) brittle and ductile materials.
model may be required. The constitutive relations shown in Figure 4.3 provide stress-strain relations for a broad range of materials with item (c) best fitting the rock’s behavior. Real rocks have anisotropic properties and often behave in a nonlinear, elastic fashion, with time-dependent creep and elastoplastic deformation. When modeling rocks, however, we do not know all material parameters, and therefore usually assume that the rocks are linear elastic, isotropic and homogeneous. If any of the real rock properties are introduced, the mathematical formulation becomes complex. These depend largely upon the accuracy of the laboratory measurements and the subsequent analysis. Note 4.1: A key point in consistency is that, if a simplified linear elastic relation is used for rock mechanics analysis, the same relation must be applied when using the resulting stresses to develop a failure/fracture analysis.
Theory of Elasticity
47
Assuming a linear elastic model, the constitutive relation for a threedimensional structure, is given, in the x direction, by: «x ¼
1 n n sx sy sz E E E
ð4:7Þ
where the strain «x in the x direction is caused by the stresses from the three orthogonal directions. Similar expressions can be given for strains in y and z directions and also for shear strains. The constitutive relations can then be grouped into two matrices for normal strains/stresses: 2 3 2 32 3 «x 1 n n sx 1 4 «y 5 ¼ 4 n 1 n 54 sy 5 E «z sz n n 1 or ½« ¼
1 ½K ½s E
ð4:8Þ
Using the same method, Hooke’s law in shear, i.e. Equation 4.4, can be developed to represent a three-dimensional material, as given below: 2 3 2 3 g xy t xy 1 4 g yz 5 ¼ 4 t yz 5 ð4:9Þ G g xz t xz Where the shear modulus G is related to the modulus of elasticity E, by: G¼
E 2ð1 þ nÞ
ð4:10Þ
Multiplying both sides of Equation 4.8 with the inverse of [K], the constitutive relation can be expressed by: ½s ¼ E ½K 1 ½« Solving for the inverse of the [K], the stresses can be found by the following equation: 2 3 2 32 3 sx 1n n n «x E 4 sy 5 ¼ 4 n 54 «y 5 ð4:11Þ 1n n ð1 þ nÞð1 2nÞ sz «z n n 1n
48
Constitutive Relation for Rocks
Note 4.2: The three-dimensional constitutive relation is valid for any structure loaded from a relaxed state or from an initial stress state. Most rock mechanics applications have a stress state which changes from an initial stress state.
Example 4.1: A compressive stress is to be applied along the long access of a cylindrical rock plug that has a diameter of 0.4 inches. Determine the magnitude of the load required to produce a 104 inch change in diameter, if the deformation is entirely within the elastic region. Assume the module of elasticity of rock as E = 9 106 psi, and Poisson’s ratio as v = 0.25. Solution: Figure 4.4 displays the deformation of the solid rock in two directions; i.e. shortening in the y axis and expanding in the x axis (increase in diameter by 104 inches), i.e.: «x ¼
Dd di do 104 in in m in ¼ ¼ 2:5 104 ¼ 250 ¼ d 0:4 in in in do
where m is Micron measure equal to 106 To calculate the strain in y direction, we use Equation 4.3, i.e. «y ¼
«x 2:5 104 in m in ¼ ¼ 103 ¼ 1000 in in v 0:25
[(Figure_4)TD$IG]
Figure 4.4 Linear deformation of a solid rock under axial compression.
Theory of Elasticity
49
This indicates that the solid rock’s axial reduction is four times larger than its lateral expansion. The applied stress can now be calculated using Hooke’s law, i.e. Equation 4.1, as: in ¼ 9000 psi ¼ 9 ksi sy ¼ E«y ¼ 9 106 psi 0:001 in
where ksi = 103 psi. Finally the applied force can be calculated as F ¼ sy Ao , where Ao, for a solid circular rock rod, is Ao ¼
pd2o p 0:42 in2 ¼ ¼ 0:126 in2 4 4
and therefore the resulting compressive load is: F ¼ sy Ao ¼ 9000 psi 0:126 in2 ¼ 9000
lbf 0:126 in2 ¼ 1130:98 lbf D1131lbf in2
Problems 4.1: A vertical concrete pipe of length L = 1.0 m, outside diameter dout = 15 cm, inside diameter din = 10 cm is compressed by an axial force F = 1.2 kN as shown in Figure 4.5. Assuming the modulus of elasticity of concrete to be E = 25 GPa and Poisson’s ratio as v = 0.18, determine: (a) the shortening in the pipe length (b) the lateral strain «y
[(Figure_5)TD$IG]
Figure 4.5 A concrete pipe under axial compression.
50
Constitutive Relation for Rocks
(c) The increase in the outer and inner diameters Ddout and Ddin (d) The increase in the pipe wall thickness Dt (e) The percentage change in the volume of the tube (DV / V) 100. 4.2: Using the concept of scientific and engineering strains defined in Section 3.3, show that the following relationship exists between the two strain definitions: «i ¼
« 1þ«
Assuming a constant E, perform a similar analysis for respective stresses and show that: si ¼ s
li 1 ¼s 1þ« lo
Finally, given the elongation of a bar as: Dl ¼
Fl Fli ¼ EA EAi
Table 4.1 Force length test data
Force (kN) Length (mm)
0 50
11 501
13 50.2
18 50.3
and assuming E = 70 MPa, lo = 50 mm, d = 10 mm, and test data given above, compute both strains, and plot the respective stresses. 4.3: The stress-strain relation for a sample rock tested in the laboratory is shown schematically in Figure 4.6. It is intended to simplify this non-linear relation by an elastic/inelastic model to best fit the material behavior. During the laboratory test, the
[(Figure_6)TD$IG]
Figure 4.6 Stress-strain relation for a laboratory test rock sample.
Theory of Elasticity
51
modulus of elasticity and yield strain were estimated as 75 GPa and 0.25%, respectively. The material strain at the point of failure was also measured as to be 2.7%. Determine: (a) Yield strength and strain hardening factor (b) Develop linear elastic and inelastic equations representing this rock sample, plot the liner models in scale and discuss the accuracy of the results.
CHAPTER
Theory of Elasticity 4.1 INTRODUCTION Many engineering components, while in service, are subjected to a variety of loading. Examples include an airplane wing, which is subjected to lift and drag forces and formation rocks, which are subjected to pore pressure, in-situ stresses and the forces imposed by a drilling bit. In such situations it is necessary to know the characteristics of the engineering component’s materials such that any resulting deformation will not be excessive and failure will not occur. This is achieved by means of stress-strain relationships. The theory of elasticity a methodology that creates a linear relation between the imposing force (stress) and resulting deformation (strain), for the majority of materials which behave fully or partially elastically. It plays an important role in the design of man-made components and structures as well as the assessment of the integrity of stable natural systems that have been disturbed by man. The principles of the theory of elasticity and relevant stress-strain equations are discussed in this chapter, as prerequisites to studying failure criteria in Chapter 5.
4.2 MATERIALS BEHAVIOR The degree to which a structure strains depends on the magnitude of an imposed load or stress. The element of stress is therefore very central in solids mechanics. However, stress cannot be measured directly. Usually strain (deformation) is measured in-situ or in the laboratory, and the stresses are then calculated. For many materials that are stressed in tension or compression at very low levels, stress and strain are proportional to each other through a simple linear relationship as shown in part (a) of Figure 4.1. However, the stress-strain (force-deformation) relation is not always simple and linear, and it can change with change of material properties and
Petroleum Rock Mechanics DOI 10.1016/B978-0-12-385546-6.00004-8
2011 Elsevier Ltd. All rights reserved.
|
41
42
Materials Behavior
[(Figure_1)TD$IG]
Figure 4.1 Typical stress-strain behavior of a material from onset of loading to fracture: (a) elastic deformation, (b) early plastic deformation, (c) further plastic deformation, (d) failure due to plastic deformation. (Callister, 2000).
geometry. Stress-strain relations, known also as constitutive relations, are normally found empirically. Consider a rod with initial length lo in tension, as shown in Figure 4.2a. Having applied an axial load to the rod, it stretches to an additional length Dl = l lo. It can be seen that the rod elongates in the axial direction but to minimize volume increase, it will become thinner in the middle section.
[(Figure_2)TD$IG]
Figure 4.2 Stress-strain diagram showing linear elastic deformation.
Theory of Elasticity
43
4.3 HOOKE’S LAW A linear relationship exists between the stress and strain as shown in Figure 4.2b and is given by the following equation: sx ¼ E«x
ð4:1Þ
where: sx ¼
Fx A
and the engineering strain definition of «x, i.e. Equation 1.4, is used: «x ¼
Dl lo
Equation 4.1 is known as Hooke’s law of deformation, and the slope of the stress-strain diagram is referred to as the Modulus of Elasticity (Young’s modulus), E. Using sx and «x in Equation 4.1, Hooke’s law can also be expressed as: Dl ¼
F x lo EA
ð4:2Þ
The tensor representation of Hooke’s law is given by: sij ¼ L ijkl «kl
ð4:3Þ
where all stress and strain components are coupled with anisotropic properties. It should be noted that the simplest possible representation should always be used, as the above equation can be very complex and difficult to solve. The ratio of traverse strain «y to axial strain «x is defined as the Poisson’s ratio, and is expressed by: n¼
«y «x
ð4:4Þ
4.4 HOOKE’S LAW IN SHEAR The properties of materials in shear can be evaluated from shear tests or by other methods such as a torsion test. In either method, a plot of shear stress
44
Analysis of Structures
versus shear strain is produced, which is similar to the strain-stress plot that results from a normal tension test (see Figure 4.1). This is known as shear stress-shear strain relation. From the stress-strain plot, we can determine properties such as the modulus of elasticity, yield strength and ultimate strength. These properties in shear are usually about half the size of similar properties in tension, and they are evaluated using a shear stress-shear strain plot. For many materials, the initial part of a shear stress-shear strain plot is a straight line, which is the same as the plot that results from a tension test. For this linearly elastic region, the shear stress and shear strain are proportional, and can therefore be expressed in a similar form to Equation 4.1 as given below: t ¼ Gg
ð4:5Þ
where t is the shear stress, G is shear modulus or modulus of rigidity and g is the shear strain. Equation 4.5 is the one dimensional Hooke’s law in shear.
4.5 ANALYSIS OF STRUCTURES Structures can be categorized into two main groups: (i) statically determinate structures and (ii) statically indeterminate structures. The first group’s feature is that their reactions and internal forces can be determined solely from free-body diagrams and the equation of equilibrium. The second group of structures is, however, more complex, and their reactions and internal forces cannot be evaluated by the equation of equilibrium alone. To analyze such structures we must provide additional equations pertaining to its deformation/displacement. The following equations must be satisfied by a statically indeterminate structure and have to be solved simultaneously: Equation of Equilibrium: This equation results from a free body diagram where a relationship exists between applied forces, reactions and internal forces. Equation of Compatibility: This equation expresses the fact that the changes in dimensions must be compatible with the boundary conditions. Constitutive Relation (Stress-Strain Equation): This relation expresses the link between acting forces (stresses) and deformations/displacements (strains). This relation, as explained earlier, has various forms depending upon the properties of material.
Theory of Elasticity
45
4.6 THEORY OF INELASTICITY Referring to the definition of an elastic body as one in which the strain at any point on the body is linearly related to and completely determined by stress, then an obvious definition of an inelastic body is one in which the stress-strain relationship is not linear due to additional elements which may affect the material’s behavior. Theory of inelasticity is complex, and very much dependent on the behavior of the material. However, for materials that have two rather distinct elastic and inelastic regions, separated by a yield strength point, the theory of inelasticity can be simplified by using a continuous function to approximate the stress-strain relation over both the elastic and inelastic regions. In this model, the stress strain relation is approximated by two straight lines, one describing slope E as defined by Hooke’s law in Equation 4.1, and the other with slope aE as given below: sx ¼ ð1 aÞS y þ aE«x
ð4:6Þ
where Sy is the yield strength, E is the modulus of elasticity and a is the strain hardening factor. The linear elastic and inelastic relation offers many advantages, especially because it results in an explicit mathematical formulation and can therefore be used for most applications. More complex, material formulations are introduced when certain applications are required. These relations are dependent strongly upon the deformation of the material.
4.7 CONSTITUTIVE RELATION FOR ROCKS In Section 4.2, a linear stress-strain model was shown as being a usual case for metals. It should however be noted that at high loads, metals may yield before failure, and for some applications a more accurate relation may be required. Therefore, in Section 4.6 an additional, simplified, linear stress-strain model was developed to account for the inelastic behavior of a metal. Rocks behave similarly to brittle metals; i.e. they deform linearly at small loads, but in a non-linear or plastic fashion at higher loads. Therefore, the combined, linearized elastic and inelastic model may represent many materials, including rocks, with a reasonable accuracy. However, depending on the rock materials, sometimes a more accurate
46
Constitutive Relation for Rocks
[(Figure_3)TD$IG]
Figure 4.3 Schematic plots of different constitutive relationships; (a) perfect elastic material, (b) real material in tension, (c) elastoplastic material, (d) brittle and ductile materials.
model may be required. The constitutive relations shown in Figure 4.3 provide stress-strain relations for a broad range of materials with item (c) best fitting the rock’s behavior. Real rocks have anisotropic properties and often behave in a nonlinear, elastic fashion, with time-dependent creep and elastoplastic deformation. When modeling rocks, however, we do not know all material parameters, and therefore usually assume that the rocks are linear elastic, isotropic and homogeneous. If any of the real rock properties are introduced, the mathematical formulation becomes complex. These depend largely upon the accuracy of the laboratory measurements and the subsequent analysis. Note 4.1: A key point in consistency is that, if a simplified linear elastic relation is used for rock mechanics analysis, the same relation must be applied when using the resulting stresses to develop a failure/fracture analysis.
Theory of Elasticity
47
Assuming a linear elastic model, the constitutive relation for a threedimensional structure, is given, in the x direction, by: «x ¼
1 n n sx sy sz E E E
ð4:7Þ
where the strain «x in the x direction is caused by the stresses from the three orthogonal directions. Similar expressions can be given for strains in y and z directions and also for shear strains. The constitutive relations can then be grouped into two matrices for normal strains/stresses: 2 3 2 32 3 «x 1 n n sx 1 4 «y 5 ¼ 4 n 1 n 54 sy 5 E «z sz n n 1 or ½« ¼
1 ½K ½s E
ð4:8Þ
Using the same method, Hooke’s law in shear, i.e. Equation 4.4, can be developed to represent a three-dimensional material, as given below: 2 3 2 3 g xy t xy 1 4 g yz 5 ¼ 4 t yz 5 ð4:9Þ G g xz t xz Where the shear modulus G is related to the modulus of elasticity E, by: G¼
E 2ð1 þ nÞ
ð4:10Þ
Multiplying both sides of Equation 4.8 with the inverse of [K], the constitutive relation can be expressed by: ½s ¼ E ½K 1 ½« Solving for the inverse of the [K], the stresses can be found by the following equation: 2 3 2 32 3 sx 1n n n «x E 4 sy 5 ¼ 4 n 54 «y 5 ð4:11Þ 1n n ð1 þ nÞð1 2nÞ sz «z n n 1n
48
Constitutive Relation for Rocks
Note 4.2: The three-dimensional constitutive relation is valid for any structure loaded from a relaxed state or from an initial stress state. Most rock mechanics applications have a stress state which changes from an initial stress state.
Example 4.1: A compressive stress is to be applied along the long access of a cylindrical rock plug that has a diameter of 0.4 inches. Determine the magnitude of the load required to produce a 104 inch change in diameter, if the deformation is entirely within the elastic region. Assume the module of elasticity of rock as E = 9 106 psi, and Poisson’s ratio as v = 0.25. Solution: Figure 4.4 displays the deformation of the solid rock in two directions; i.e. shortening in the y axis and expanding in the x axis (increase in diameter by 104 inches), i.e.: «x ¼
Dd di do 104 in in m in ¼ ¼ 2:5 104 ¼ 250 ¼ d 0:4 in in in do
where m is Micron measure equal to 106 To calculate the strain in y direction, we use Equation 4.3, i.e. «y ¼
«x 2:5 104 in m in ¼ ¼ 103 ¼ 1000 in in v 0:25
[(Figure_4)TD$IG]
Figure 4.4 Linear deformation of a solid rock under axial compression.
Theory of Elasticity
49
This indicates that the solid rock’s axial reduction is four times larger than its lateral expansion. The applied stress can now be calculated using Hooke’s law, i.e. Equation 4.1, as: in ¼ 9000 psi ¼ 9 ksi sy ¼ E«y ¼ 9 106 psi 0:001 in
where ksi = 103 psi. Finally the applied force can be calculated as F ¼ sy Ao , where Ao, for a solid circular rock rod, is Ao ¼
pd2o p 0:42 in2 ¼ ¼ 0:126 in2 4 4
and therefore the resulting compressive load is: F ¼ sy Ao ¼ 9000 psi 0:126 in2 ¼ 9000
lbf 0:126 in2 ¼ 1130:98 lbf D1131lbf in2
Problems 4.1: A vertical concrete pipe of length L = 1.0 m, outside diameter dout = 15 cm, inside diameter din = 10 cm is compressed by an axial force F = 1.2 kN as shown in Figure 4.5. Assuming the modulus of elasticity of concrete to be E = 25 GPa and Poisson’s ratio as v = 0.18, determine: (a) the shortening in the pipe length (b) the lateral strain «y
[(Figure_5)TD$IG]
Figure 4.5 A concrete pipe under axial compression.
50
Constitutive Relation for Rocks
(c) The increase in the outer and inner diameters Ddout and Ddin (d) The increase in the pipe wall thickness Dt (e) The percentage change in the volume of the tube (DV / V) 100. 4.2: Using the concept of scientific and engineering strains defined in Section 3.3, show that the following relationship exists between the two strain definitions: «i ¼
« 1þ«
Assuming a constant E, perform a similar analysis for respective stresses and show that: si ¼ s
li 1 ¼s 1þ« lo
Finally, given the elongation of a bar as: Dl ¼
Fl Fli ¼ EA EAi
Table 4.1 Force length test data
Force (kN) Length (mm)
0 50
11 501
13 50.2
18 50.3
and assuming E = 70 MPa, lo = 50 mm, d = 10 mm, and test data given above, compute both strains, and plot the respective stresses. 4.3: The stress-strain relation for a sample rock tested in the laboratory is shown schematically in Figure 4.6. It is intended to simplify this non-linear relation by an elastic/inelastic model to best fit the material behavior. During the laboratory test, the
[(Figure_6)TD$IG]
Figure 4.6 Stress-strain relation for a laboratory test rock sample.
Theory of Elasticity
51
modulus of elasticity and yield strain were estimated as 75 GPa and 0.25%, respectively. The material strain at the point of failure was also measured as to be 2.7%. Determine: (a) Yield strength and strain hardening factor (b) Develop linear elastic and inelastic equations representing this rock sample, plot the liner models in scale and discuss the accuracy of the results.