3-2.ppt

College of Materials Science & Engineering
Shear modulus
NORTH UNIVERSITY OF CHINA Shear stress and strain are proportional to each other through the expression:
of a secant drawn from the origin to some given point of the stress-strain curve.
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Microscopic understanding for E
College of Materials Science & Engineering
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Polymers have modulus values that are smaller than both metals and ceramics, and which lie in the range 0.007 and 4 GPa.
Deformation is reversible, nonpermanent.
College of Materials Science & Engineering
Stress-Strain Behavior
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Hooke’s law: It shows that stress and strain are proportional to each other. The constant of proportionality E (GPa) is the modulus of elasticity, or Young's modulus. How to understand the meaning of E:
、正割模数、视模数 ） represents the slope
Figure: Schematic stress-strain diagram showing nonlinear elastic behavior, and how secant (割线) and tangent (切线) moduli are determined.
Tangent modulus is taken as the slope of the stress-strain curve at some specified level of stress, while secant modulus （ 割 线 模 量
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College of Materials Science & Engineering
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Linear elastic behavior: A plot of stress versus strain results in a linear relationship, as shown in the left figure. The slope of this linear segment corresponds to the modulus of elasticity E.
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Solution:
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since the deformation is elastic, strain is dependent on stress according to σ = Eε; furthermore, the elongation Δl is related to the original length l0 through ε=(li-l0)/l0=Δl/l0, combing these two expressions and solving for Δl yields: σ=εE=(Δl/l0 )E Δl=σ l0/ E The values of σ and l0 are given in the question, by substitution of the given value, Δl=0.77 mm.
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Figure: Force versus interatomic separation for weakly and strongly bonded atoms. The magnitude of the modulus of elasticity is proportional to the slope of each curve at the equilibrium interatomic separation.
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The magnitude of E is a measure of the interatomic bonding forces; this modulus is proportional to the slope of the interatomic force-separation curve at the equilibrium spacing (平衡位置):
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Time-dependent elastic behavior The behavior is elastic but the stress-strain curve is not immediately reversible. The elastic deformation will continue after the stress application, and upon load release some finite time is required for complete recovery.
Example problem：
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A piece of copper originally 305 mm long is pulled in tension with a stress of 276 MPa. If the deformation is entirely elastics, what will be the resultant elongation?
It may be thought of as stiffness (刚度), or a material’s resistance to elastic deformation. The greater the modulus, the stiffer the material, or the smaller the elastic strain that results from the application of a given stress.
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Characteristic of E
NORTH UNIVERSITY OF CHINA With increasing temperature, E diminishes for all but some of the rubber materials.
College of Materials Science & Engineering
For example:
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For most typical metals the magnitude of the modulus ranges between 45 GPa and 407 GPa
3.3 Elastic Deformation (弹性变形)
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Elastic deformation: When the stress is removed, the material returns to the dimension it had before the load was applied. Valid for small strains (0.5~1％) (except the case of rubbers).
College of Materials Science & Engineering
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定义：材料在弹性变形阶段，其应力和应变成正比例关系（即符合 胡克定律），其比例系数称为弹性模量。 单位：达因每平方厘米。 意义：弹性模量可视为衡量材料产生弹性变形难易程度的指标，其 值越大，使材料发生一定弹性变形的应力也越大，即材料刚度越大， 亦即在一定应力作用下，发生弹性变形越小 说明:又称杨氏模量。弹性材料的一种最重要、最具特征的力学性质。 是物体弹性t变形难易程度的表征。用E表示。定义为理想材料有小 形变时应力与相应的应变之比。E以单位面积上承受的力表示，单 位为牛/米^2。模量的性质依赖于形变的性质。剪切形变时的模量称 为剪切模量，用G表示；压缩形变时的模量称为压缩模量，用K表 示。模量的倒数称为柔量，用J表示。
College of Materials Science & Engineering
NORTH UNIVERSITY OF CHINA
The effect is normally small for metals but can be significant for polymers.
College of Materials Science & Engineering
College of Materials Science & Engineering
NORTH UNIVERSITY OF CHINA
刚度是指在产生单位位移时所需要的力,刚度表 明构件的抵抗变形的能力。 刚度是指零件在载荷作用下抵抗弹性变形的能力。 零件的刚度（或称刚性）常用单位变形所需的力 或力矩来表示，刚度的大小取决于零件的几何形 状和材料种类（即材料的弹性模量）。刚度要求 对于某些弹性变形量超过一定数值后，会影响机 器工作质量的零件尤为重要，如机床的主轴、导 轨、丝杠等。
Gpa
psi
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The moduli of elasticity are slightly higher for ceramic materials, which range between about 70 and 500 GPa
Differences of E are a direct consequence of the different types of atomic bonding that exist for the different materials types.
College of Materials Science & Engineering
College of Materials Science & Engineering
Figure: Schematic stressstrain diagram showing linear elastic deformation for loading and unloading cycles.
Nonlinear elastic behavior
where G is the shear modulus, the slope of the linear region of the shear stress-stain curves.
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Anelasticity (粘弹性)