(3) Stress at a Point and Stress Field

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Lecture 3 Stress at a Point and Stress Field 第三讲 一点的应力和应力场

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Stress at a Point and Stress Field Force力，surface forces面力，body force体力 External forces外力，Internal forces内力，stresses应力 Normal stress and shear stress正应力与剪应力 Stress components(分量 )in three dimensions Principal stresses(主应力), principal planes(主平面, principal axes(主轴) and principal directions(主方向) Stress ellipse and stress ellipsoid 应力椭圆和应力椭球 Classes of stress states Stress field 应力场

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Force力Force is a kind of mechanical action between different objects, it tends to change the shape, volume or movement state of the object with a force upon it.Force is a vector quantity, and thus possesses both magnitude and direction; it can be represented by an arrow whose length specifies the magnitude and whose orientation specifies the orientation of the force. Force = mass × acceleration (kg m s-2) [Newton][N]

F vector

Unit: Newton 1 Newton = 1 kilogram meter per second squared scalar (only magnitude)

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F1

F

F1 F F2 B

F2 A

Resolution and resultant of forces 力的分解与合成 A.A force F be resolved into two components F1 and F2. B. Two forces F1 and F2 be represented by the resultant F

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Surface Forces and Body Forces 面力与体力Surface forces: the forces acting on the contact surface between adjacent parts of rock system, between adjacent blocks or adjacent lithosphere plates. The contact surface may be or may be not a visible material boundary. It can be a imaginary surface inside the object considered.

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Body forcesBody forces: the forces can work at a distance and depend on the amount of material affected, so, we can call body forces distant forces. Gravitational force is an example of body forces. The gravitational force on a rock body of mass m is F = mg where g is the acceleration of gravity. g varies with depth in the earth and with position on the earth’s surface, but for the purpose of structural geology, it is a constant 9.8m/sec2.

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Nonuniform forces F a b Imaginary plane x N=F a σ Uniform Internal Forces F

Uniform forces

F

σ=N/A=F/A

External forces 外力

Internal forces and stresses 内力与应力

Stress on a plane: internal forces acting on unit area of the given plane within the considering body.

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External forces

P m F

P—Internalfo rc e Inte rna lforces

F—aarea re a

stress

P P dp lim F = dF = p F →0

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Normal stress and shear stress正应力 正应力 与剪应力

Stress acting at a point m on a plane n is a vector, it can be resolved into two components σ and τ, σ is normal to the plane, called normal stress, τ is tangential to the plane, called shear stress.

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Magnitude and Units of StressMagnitude of stress Stress = Force / Area, limit Area approaching zero Units of stress [ Newton / m2 or a ‘Pascal’], or simply say ‘Pa’ That is 1 pascal = 1 newton per

square meter. 1 newton = 1 kilogram meter per second squared (1 kg m s-2) A more commonly used unit is the bar or the kilobar, Where: 1 bar = 105 pascals = 0.1 Mpa (105: 10 to the power 5)

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Normal and shear stresses at a fault plane (A) and a bedding plane during flexural slip folding (B) (Park,1997)

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τyp p pz y x

τx

Resolution of stress in two dimensions (A) and in three dimensions (B)(Park,1997)

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τxy=τyx τyz=τzy τzx=τxz

Infinitesimal cube

Stress components(分量 )in three ( dimensions

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Nine components of stress at a point in matrix form

(5-2)τxy=τyx τyz=τzy τzx=τxz , only six stress components left: three normal stresses σx, σy ,σz and three shear stresses τxy , τy z, τzxSince For an arbitrarily chosen set of orthogonal axes x, y, z, six independent quantities are necessary to specify completely the state of stress at a point.

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Principal stresses(主应力), principal planes(主 (主应力) ( 平面, 平面 principal Axes(主轴) and principal (主轴) directions(主方向) (主方向)The components vary as the orientation of the orthogonal coordinate axes change. It can be proved that at any point we can find an orientation for the cube such that the shear stresses on the surfaces of the cube are all zero. That is:

τxy = τy z = τzx = 0

The only three normal stresses left are called principal stresses. The three mutually perpendicular planes on which the shear stress is zero are called principal stress planes. The axes are called principal stress axes. The directions along the principal stress axes are called principal stress directions.

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Greatest, intermediate and least principal stressesThe conventional notation for principal stresses are σ1 , σ2 , σ3, where σ1 ≥ σ2 ≥ σ3, or called greatest, intermediate and least principal stresses, respectively. If the three principal stresses at a point are determined, the state of stress at the point is completely specified.

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中面(middle plane)

Middle plane 中面

Thin plate薄板

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Stress ellipse defined by tails of stress vectors acting across various planes at a point p. The two stress vectors σAB and σBA associated with a particular plane p are identified. (after W.D. Means, 1976)

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Stress ellipse defined by heads of stress vectors across various planes at a point. In this case the normal component of each stress is tensile (not compressive). (after W.D. Means,1976)

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σ σ σ1

1

σ1> σ2 > σ3σ2

3

(A)

σ σ2

1

σ(B)

σ3

3

σ

2

(A) General triaxial stress ellipsoid in perspective view.(B)Views normal to each of the principal planes of the ellipsoid. (After W.D. Means,1976)

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Classes of Stress States

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