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實驗力學研究室 Fundamentals

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**Body Under External Loading**

實驗力學研究室 First Principles Body Under External Loading First Law: A body will remain at rest or will continue its straight line motion with constant velocity if there is no unbalanced force acting on it. Second Law: the acceleration of a body will be proportional to the resultant of all forces acting on it and in the direction of the resultant. Third Law: Action and reaction forces between interacting bodies will be equal in magnitude, collinear, and opposite in direction.

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**G, the center of gravity of the body.**

實驗力學研究室 G, the center of gravity of the body.

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**G= mv constitutes the linear momentum vector of a body.**

實驗力學研究室 G= mv constitutes the linear momentum vector of a body. H is the angular momentum vector of the body.

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**Constraining the body to uniaxial motion,**

實驗力學研究室 Constraining the body to uniaxial motion, Constraining the body to planar motion, aG is the vectorial acceleration of the center of gravity (c.g.) of the body. IG is the mass moment of inertia of the body about an axis normal to the plane of motion through the c.g., and α is the body’s angular acceleration.

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**Stress and Strain What Is Stress?**

實驗力學研究室 Stress and Strain What Is Stress? For static equilibrium, τxy=τyx, τyz=τzy, and τzx=τxz.

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實驗力學研究室 Principal Stresses In a given loaded structure, a particular element orientation exists for which all the shear stress components are zero. The normals to the faces of an element in this orientation are called principal directions and the stresses along these normals are the principal stresses.when one of the principal stresses is zero, the stress state is considered to be biaxial or plane stress. These problems can be deconstructed into planar approximations in which the loading and boundary conditions are in that plane and identical on any parallel plane.

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**Maximum/minimum Principal Stresses**

實驗力學研究室 Plane Stress Case: Maximum/minimum Principal Stresses Maximum/minimum Shear Stresses (45° away from the orientation of principal stress)

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實驗力學研究室 Triaxial Case: where I1, I2, and I3 are called stress invariants. I1, or the first invariant, is the internal hydrostatic pressure. The maximum shear stress is given in terms of the maximum and minimum principal stresses as follows.

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實驗力學研究室 Strain

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實驗力學研究室 Principal Strain The strains that occur in the direction of principal stresses are known as principal strains.

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**Fundamental Stress States**

實驗力學研究室 Fundamental Stress States Stress in Flexure

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實驗力學研究室 Stress in shear where V is the shear force, b is the width of the stress section, and Q is the first moment of area of the section about a transverse axis with origin on the neutral axis.

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實驗力學研究室 Stress in Torsion Here, r is the redius from the torsional axis and J is the section’s polar area moment of inertia.

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**where G is the material’s modulus of rigidity. **

實驗力學研究室 where G is the material’s modulus of rigidity. The approximate formula for a rectangular section beam of width (w) and thickness (t).

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實驗力學研究室 Stress in Pressure

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實驗力學研究室 where σt is also known as the hoop stress in the cylinder, σr is radial stress. The longitudinal stress due to pressure on the end caps is constant throughout the cylinder and is given by the equation.

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實驗力學研究室 If the pressure created due to an equal-length cylindrical press-fit was known, above equations could be used for obtaining the stress state on both the outside and inside cylinders. where ri is the inside cylinder’s internal radius, ro is the outside cylinder’s external radius, R is the transition radius, and Ei, Eo, vi, and vo are the inside and outside material Young’s moduli and Poisson’s ratios, respectively. δ is the radial interference.

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**Stress in Contact (Hertzian stresses )**

實驗力學研究室 Stress in Contact (Hertzian stresses ) Contact circular area resulting from the forced contact of two spheres will be

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實驗力學研究室 At the center of this area, a maximum pressure pmax will occur of the following magnitude. For two cylinders of equal length l and diameters d1 and d2, the resulting contact surface is a rectangle of length l and width 2b, where The maximum pressure occurs along the long center line of the rectangle.

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**Stress in thermal Expansion**

實驗力學研究室 Stress in thermal Expansion The constant of proportionality α is know as the material’s coefficient of thermal expansion. For a straight beam constrained at both ends, the resulting compressive stress at a distance from the ends is given by the next equation.

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**Stress Concentration Factors **

實驗力學研究室 Stress Concentration Factors The stress concentrations will also appear close to unplanned irregularities in the part, such as cracks and pits. σo and τo are the nominal stresses found in the part without the feature.

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**Material Properties Types of Materials**

實驗力學研究室 Material Properties Types of Materials Isotropic. Properties are the same in any direction or at any cross section. Anisotropic. Properties differ in two or more directions. Orthotropic. Specific type of anisotropic in which planes of extreme values are orthogonal(i.e.,perpendicular to one another).

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實驗力學研究室 Common Material Properties (modulus of elasticity (E), modulus of rigidity (G), and Poisson’s ratio (v))

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實驗力學研究室 Point P, known as the proportional limit. Point E, the elastic limit. Point Y is the yield point of the material, corresponding to its yield strength (Sy). Point U indicates the maximum stress that can be achieved by the material. This corresponds to its ultimate or tensile strength. Fracture point (F), which marks the fracture strength (SF) of the material.

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**Ductile versus Brittle Material Behavior**

實驗力學研究室 Ductile versus Brittle Material Behavior If permanent set (plastic deformation) is obtainable, the material is said to exhibit ductility. For ductile materials , the ultimate tensile and compressive strengths have approximately the same absolute value. Brittle materials on the other hand are stronger in compression than in tension.

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**Brittle materials exhibit the behavior described below. **

實驗力學研究室 Brittle materials exhibit the behavior described below. A graph of stress versus strain is a smooth, elastic curve until failure which manifests as fracture. Materials behaving in this manner do not have a “yield strength.” Compressive strength is usually many times greater than tensile strength. Modulus of rupture is approximately the same as tensile strength. Rapid crack propagation along cleavage planes occurs with no noticeable plastic deformation.

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**If the percent elongation is at or below 5%, assume brittle behavior. **

實驗力學研究室 Rules of thumb used to determine if brittle or ductile behavior should be expected are summarized below. If the percent elongation is at or below 5%, assume brittle behavior. If the published ultimate compressive strength is greater than the ultimate tensile strength, assume brittle behavior. If no yield strength is published, suspect brittle behavior.

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**Failure Modes Typical Failure Modes**

實驗力學研究室 Failure Modes Typical Failure Modes Fracture. Fracture is said to occur when new cracks appear or existing cracks are extended. A brittle fracture is one that exhibits little or no permanent (plastic) deformation. Yielding. A body which experiences stresses in excess of the yield strength is said to have failed only when this yielding compromises the integrity or function of the part. Yielding near stress concentrations is not considered a failure if it produces localized strains which merely redistribute the stress, whereupon yielding ceases.

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實驗力學研究室 Insufficient stiffness. Parts must be stiff enough to hold tolerances and support required loads. Moving parts may have undesirable resonant frequencies if they are too flexible. Buckling. The sudden loss of stability or stiffness under applied load. Stress levels need not be high for buckling to occur. Fatigue. Parts that are subject to variable loading will lose strength with time and may fail after a certain number of cycles. Creep. Bodies under load gradually deform over time. The apparent modulus property is derived form empirical creep data for various materials and may be used to compensate for the effects of creep.

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**Classic Failure theories**

實驗力學研究室 Classic Failure theories Ductile Failure theory Maximum normal stress theory. Failure occurs whenever σ1 or σ3 equals the failure strength of the material in tension or compression, respectively. Maximum shear stress theory ( Tresca criterion ). Yielding begins when the maximum shear stress becomes equal to one-half the yield strength. Failure in tension of ductile materials occurs on one of the 45°maximum shear planes. Annealed ductile materials tend to fail according to this theory.

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實驗力學研究室 Distortion energy ( Von Mises-Hencky ) theory (Suitable for entire stress state). Probably the most widely used, this theory predicts that failure by yielding will occur whenever the von Mises, or effective stress (σ’ ), equals the yield strength of the materials.

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**Brittle Failure Theory **

實驗力學研究室 Brittle Failure Theory Maximum normal stress. Similar to that defined for ductile materials. Failure occurs when the ultimate strength, not yield, is reached. Coulomb-Mohr theory. Fracture occurs when the maximum and minimum principal stresses combine for a condition which satisfies the following: where Sut and Suc represent the ultimate tensile and compressive strengths, and both σ3 and Suc are always negative, or in compression.

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實驗力學研究室 Modified Mohr theory. Fracture occurs as defined in the Coulomb-Mohr theory except in the fourth quadrant condition where σ1 is in tension and σ2 is in compression.

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**Other Failure Theories**

實驗力學研究室 Other Failure Theories Buckling where E is the modulus of elasticity of the column’s materials, I is the smallest or least moment of inertia of its cross-sectional area, and Le is its effective length. The last term, Le=KL, depends on the actual length L of the column and an effective length factor K, which is assigned according to the constraint conditions of the column ends.

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**If a slenderness ratio (Le/r) is defined as**

實驗力學研究室 r as the smallest radius of gyration of the column’s cross-sectional area (A). A corresponding critical stress(σcr) may be calculated as seen in the next equation. If a slenderness ratio (Le/r) is defined as the column is considered Euler, and a critical load must be calculated and recorded.

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實驗力學研究室 For nonEuler columns, a tangent modulus variable (Et) has taken the place of the elastic modulus (function of location), below the yield point, Et=E as expected.

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實驗力學研究室 Fatigue

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實驗力學研究室 Endurance or fatigue limit (Se) is defined as the maximum cyclic stress which a part can sustain for an “ infinite” number for cycles. The endurance limit of the actual rotating beam speciman is designated as Se'. The correlation between Se and Se' is Here, ka is a surface factor, kb is a size factor, kc is a load factor, kd is a temperature factor, and ke is an all encompassing, other miscellaneous effects factor.

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實驗力學研究室 Because nenferrous metals and alloys lack an endurance limit ( the strength of material never stablizes but keep decreasing with time). A fatigue strength (Sf’) is usually reported for 50(107) cycles of reversed stress. This strength is often as low as 1/4 Sut for some aluminum alloys.

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**where a and b are provided by**

實驗力學研究室 To obtain the fatigue strength at N cycles for a part experiencing alternating or completely reversed stress, you can curve-fit the S-N curve using the following equation: where a and b are provided by Note that Se' may be substituted for Se in above equation to predict Sf '.

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實驗力學研究室 If the completely reversed stress has an amplitude (σa), the corresponding number of cycles of life is calculated via the next equation.

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實驗力學研究室 When the mean stress (σm) is at a level other than zero, the cyclic loading is classified as fluctuating stress case. One of the most accepted equations that provides a solution to this scenario is the modified Goodman relation: where Sut is the ultimate tensile strength of the material and n is the safety factor used in the design.

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**Crack initiation. A crack begins to form within the material. **

實驗力學研究室 When a ductile material is subjected to a fatigue-type loading, there are basic structural changes that occur. In chronological order, the changes are summarized below. Crack initiation. A crack begins to form within the material. Localized crack growth. Local extrusions and intrusions occur at the surface of the part because plastic deformations are not completely reversible. Crack growth on planes of high tensile stress. The crack proceeds across the section at those points of greatest tensile stress.

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實驗力學研究室 Ultimate ductile failure. When the crack reduces the effective cross section to a size that cannot sustain the applied loads, the sample ruptures by ductile failure. All the modifying factors that affect the endurance life of a part, are summarized below. Stress concentrators. General part features as described in the “Stress and Strain” section, which cause high local stresses and thus decrease fatigue life. Surface roughness. Smooth surfaces are more crack resistant because roughness creates stress concentrators.

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實驗力學研究室 Surface conditioning. Hardening processes tend to increase fatigue strength while plating and corrosion protection tend to diminish fatigue strength. Environment. A corrosive environment greatly reduces fatigue strength. A combination of corrosive attack and cyclic stresses is called corrosion fatigue.

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實驗力學研究室 Creep

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**The stress state for a viscoelastic material can be expressed as**

實驗力學研究室 The stress state for a viscoelastic material can be expressed as The four stages of creep failure, as shown in figure, are described below. Instantaneous elongation. Normal deformation under applied load. Primary creep. Material strain hardens under load to decrease creep rate. Secondary creep. Material elongates at a steady rate, called minimum creep rate. Tertiary creep. Due to necking and formation of voids, elongation proceeds at an increasing rate until fracture.

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實驗力學研究室 The secondary phase is of significant interest to engineers because it dominates the actual creep process from a time standpoint. Creep strength is defined as the stress which produces a minimum creep rate of 10-5 % per hour.

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