chapter 5. stress-strain relationships and behavior
TRANSCRIPT
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Mechanical Strengths and Behavior of Solids
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Chapter 5. Stress-Strain Relationships and Behavior
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Contents
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1 Introduction
2 Models for Deformation Behavior
3 Elastic Deformation
4 Anisotropic Materials
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5.1 Introduction
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โข Three major types of deformation: Elastic, Plastic, and Creep deformation
โข In engineering analysis, constitutive equations which describe stress-strain relationships are essential to calculate stresses and deflections in mechanical components.
โข Consider one-dimensional stress-strain behavior and some corresponding simple physical models for each deformation.
โข Three-dimensional elastic deformation: Isotropy and Anisotropyโ Isotropy: The elastic properties are the same in all directions.โ Anisotropy: The elastic properties vary with direction.
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5.2 Models for Deformation Behavior
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Figure 5.1 Mechanical models for four types of deformation. The displacementโtime and forceโdisplacement responses are also shown for step inputs of force P, which is analogous to stress ฯ. Displacement x is analogous to strain ฮต.
๐๐ : coefficient of tensile viscosity(in analogous to c)
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Plastic Deformation Models (1)
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Figure 5.3 Rheological models for plastic deformation and their responses to three different strain inputs. Model (a) has behavior that is rigid, perfectly plastic; (b) elastic, perfectly plastic; and (c) elastic, linear hardening.
๐ธ๐ธ1๐ธ๐ธ2๐ธ๐ธ1 + ๐ธ๐ธ2
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Plastic Deformation Models (2)
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โข The point during unloading where the stress passes through zeroโ Elastic unloading of the same slope ๐ธ๐ธ1 as the initial loadingโ The elastic strain ๐๐๐๐ is recovered corresponds to the relaxation of spring ๐ธ๐ธ1.โ The permanent or plastic strain ๐๐๐๐ corresponds to the motion of the slider
up to the point of maximum strain.
Figure 5.4 Loading and unloading behavior of (a) an elastic, perfectly plastic model, (b) an elastic, linear-hardening model, and (c) a material with nonlinear hardening.
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Creep Deformation Models (1)
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Figure 5.5 Rheological models having time-dependent behavior and their responses to a stressโtime step. Both strainโtime and stressโstrain responses are shown. Model (a) exhibits steady-state creep with elastic strain added, and model (b) transient creep with elastic strain added.
Spring: ๐๐๐๐ = ๐๐โฒ
๐ธ๐ธ1
Dashpot: ๐๐๐๐ = ๐๐โฒ
๐๐1
Linear viscoelasticity Some polymers, ceramics, and metals at high temperature, but low stress
Nonlinear viscoelasticityMetals and ceramics at high stress
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Creep Deformation Models (2)
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โข For the steady-state creep model of Fig. 5.5(a),๐๐ = ๐๐๐๐ + ๐๐๐๐ =
๐๐โฒ
๐ธ๐ธ1+ ๐๐๐๐
The rate of creep strain: ๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐
= ๐๐โฒ
๐๐1
Solving ๐๐๐๐ by integration, ๐๐ = ๐๐โฒ
๐ธ๐ธ1+ ๐๐โฒ๐๐
๐๐1 After removal of the stress, elastic strain disappears, but the creep strain accumulated during 1-2 remains as a permanent strain.
โข For the transient creep model of Fig. 5.5(b),The stress in the (๐๐2,๐ธ๐ธ2) stage:
๐๐ = ๐ธ๐ธ2๐๐2 + ๐๐2 ๐๐๐๐This gives
๐๐๐๐ =๐๐๐๐๐๐๐๐๐๐ =
๐๐ โ ๐ธ๐ธ2๐๐๐๐๐๐2
Solving this differential equation,
๐๐ =๐๐โฒ
๐ธ๐ธ1+๐๐โฒ
๐ธ๐ธ2(1 โ ๐๐โ
๐ธ๐ธ2๐๐๐๐2 )
Strain rate decreases with time and creep strain asymptotically approaches the limit โ๐๐โฒ ๐ธ๐ธ2. After stress removal, the transient creep strain decreases toward zero at infinite time due to the spring. The equation for the recovery response can be obtained by solving ODE.
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Relaxation Behavior (1) (optional)
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โข Creep: Accumulation of strain with time, as under constant stressโข Recovery: Gradual disappearance of creep strain that occurs after
removal of the stressโข Relaxation: Decrease in stress when a material is held at constant strain
Figure 5.7 Stressโtime step applied to a material exhibiting strain response that includes elastic, plastic, and creep components.
Relaxation
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Relaxation Behavior (1) (optional)
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โข Creep: Accumulation of strain with time, as under constant stressโข Recovery: Gradual disappearance of creep strain that occurs after
removal of the stressโข Relaxation: Decrease in stress when a material is held at constant strain
Figure 5.6 Relaxation under constant strain for a model with steady-state creep and elastic behavior. The step in strain (a) causes stressโtime behavior as in (b), and stressโstrain behavior as in (c).
Relaxation
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Relaxation Behavior (2) (optional)
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โข About sudden strain ๐๐โฒ, it is absorbed by the spring entirely because the dashpot requires a finite time to respond.
โข Due to the total strain being held constant, the strain in the spring decreases, as the strain in the dashpot increases.
๐๐โฒ = ๐๐๐๐ + ๐๐๐๐ = ๐๐๐๐๐๐๐๐๐๐.
The stress necessary to maintain the constant strain:๐๐ = ๐ธ๐ธ1๐๐๐๐
The rate of creep strain:
๐๐๐๐ =๐๐๐๐๐๐๐๐๐๐
=๐๐๐๐1
Combining these equations and solving the differential equation:
๐๐ = ๐ธ๐ธ1๐๐โฒ๐๐โ๐ธ๐ธ1๐๐๐๐1
โข If the strain is returned to zero, the stress is forced into compression. Additional relaxation the occurs, but in the opposite direction, as the relaxation always proceeds toward zero stress.
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Elastic Constants (1)
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โข Homogeneous: A material that has the same properties at all points within the solid
โข Isotropic: A material whose properties are the same in all directions
Figure 5.8 Longitudinal extension and lateral contraction used to obtain constants for a linear-elastic material that is isotropic and homogeneous.
โ Elastic modulus๐ธ๐ธ =
๐๐๐ฅ๐ฅ๐๐๐ฅ๐ฅ
โ Poissonโs ratio
๐๐ = โ๐๐๐ก๐ก๐ก๐ก๐๐๐๐๐ก๐ก๐๐๐ก๐ก๐๐๐๐ ๐๐๐๐๐ก๐ก๐ก๐ก๐ ๐ ๐๐๐๐๐๐๐๐๐๐๐ ๐ ๐๐๐๐๐๐๐ ๐ ๐๐๐ก๐ก๐๐ ๐๐๐๐๐ก๐ก๐ก๐ก๐ ๐ ๐๐
= โ๐๐๐ฆ๐ฆ๐๐๐ฅ๐ฅ
โ From elastic modulus and Poissonโs ratio
๐๐๐ฆ๐ฆ = โ๐๐๐ธ๐ธ๐๐๐ฅ๐ฅ
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Elastic Constants (2)
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โข Poissonโs ratio is often around 0.3, 0 < ๐๐ < 0.5.
โข Negative values of ๐๐ imply lateral expansion during axial tension.
โข ๐๐ = 0.5 implies constant volume, and values lager than 0.5 imply a decrease in volume for tensile loading.
โข Small-strain theory: When dimensional changes are small, original dimensions and cross-sectional areas are used to determine stresses and strains.
โข If a given metal is alloyed with small percentages of other metals, the elastic constants ๐ธ๐ธ and ๐๐ can be approximated as being the same as the corresponding pure metal values.โ Aluminum and titanium alloys: less than 10%โ Low-alloy steels: less than 5%
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Hookeโs Law for Three Dimensions (1)
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Figure 5.9 The six components needed to completely describe the state of stress at a point.
Resulting Strain Each Direction
Stress X Y Z
๐๐๐ฅ๐ฅ ๐๐๐ฅ๐ฅ๐ธ๐ธ
โ๐๐๐๐๐ฅ๐ฅ๐ธ๐ธ
โ๐๐๐๐๐ฅ๐ฅ๐ธ๐ธ
๐๐๐ฆ๐ฆ โ๐๐๐๐๐ฆ๐ฆ๐ธ๐ธ
๐๐๐ฆ๐ฆ๐ธ๐ธ
โ๐๐๐๐๐ฆ๐ฆ๐ธ๐ธ
๐๐๐ง๐ง โ๐๐๐๐๐ง๐ง๐ธ๐ธ
โ๐๐๐๐๐ง๐ง๐ธ๐ธ
๐๐๐ง๐ง๐ธ๐ธ
๐๐๐ฅ๐ฅ =1๐ธ๐ธ๐๐๐ฅ๐ฅ โ ๐๐ ๐๐๐ฆ๐ฆ + ๐๐๐ง๐ง
๐๐๐ฆ๐ฆ =1๐ธ๐ธ
[๐๐๐ฆ๐ฆ โ ๐๐ ๐๐๐ฅ๐ฅ + ๐๐๐ง๐ง ]
๐๐๐ง๐ง =1๐ธ๐ธ
[๐๐๐ง๐ง โ ๐๐ ๐๐๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ ]
The strains caused by stresses in each direction: Total strain in each direction:
โข Three normal stresses:๐๐๐ฅ๐ฅ,๐๐๐ฆ๐ฆ , and ๐๐๐ง๐ง
โข Three shear stresses:๐๐๐ฅ๐ฅ๐ฆ๐ฆ , ๐๐๐ฆ๐ฆ๐ง๐ง, and ๐๐๐ง๐ง๐ฅ๐ฅ
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Hookeโs Law for Three Dimensions (2)
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โข Shear strains (๐บ๐บ: ๐๐๐ ๐๐๐ก๐ก๐ก๐ก ๐๐๐๐๐๐๐๐๐๐๐๐๐๐):๐พ๐พ๐ฅ๐ฅ๐ฆ๐ฆ =
๐๐๐ฅ๐ฅ๐ฆ๐ฆ๐บ๐บ
, ๐พ๐พ๐ฆ๐ฆ๐ง๐ง =๐๐๐ฆ๐ฆ๐ง๐ง๐บ๐บ
, ๐พ๐พ๐ง๐ง๐ฅ๐ฅ =๐๐๐ง๐ง๐ฅ๐ฅ๐บ๐บ
โข The shear strain on a given plane is unaffected by the shear stresses on other planes.
Figure 4.41 A round bar in torsion and the resulting state of pure shear stress and strain. The equivalent normal stresses and strains for a 45ยฐrotation of the coordinate axes are also shown.
Consider a state of pure shear stress, as in a round bar under torsion in Fig. 4.41.๐๐๐ฅ๐ฅ = ๐๐, ๐๐๐ฆ๐ฆ= โ๐๐, ๐๐๐ง๐ง= 0, ๐๐๐ฅ๐ฅ=
๐พ๐พ2
This yields
๐พ๐พ =2 1 + ๐๐
๐ธ๐ธ๐๐
From ๐บ๐บ = ๐๐/๐พ๐พ,
๐บ๐บ =๐ธ๐ธ
2(1 + ๐๐)
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Volumetric Strain
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The normal strains:
๐๐๐ฅ๐ฅ =๐๐๐๐๐๐
, ๐๐๐ฆ๐ฆ =๐๐๐๐๐๐
, ๐๐๐ง๐ง =๐๐๐๐๐๐
From ๐๐ = ๐๐๐๐๐๐,
๐๐๐๐ =๐๐๐๐๐๐๐๐
๐๐๐๐ +๐๐๐๐๐๐๐๐
๐๐๐๐ +๐๐๐๐๐๐๐๐
๐๐๐๐
Evaluating the partial derivatives and dividing both sides by ๐๐ = ๐๐๐๐๐๐,
๐๐๐๐๐๐
=๐๐๐๐๐๐
+๐๐๐๐๐๐
+๐๐๐๐๐๐
Figure 5.10 Volume change due to normal strains.
Volumetric strain or dilatation, ๐๐๐ฃ๐ฃ: Ratio of the change in volume
๐๐๐ฃ๐ฃ =๐๐๐๐๐๐
= ๐๐๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ + ๐๐๐ง๐ง
For an isotropic material (by Hookeโs law),
๐๐๐ฃ๐ฃ =1 โ 2๐๐๐ธ๐ธ
(๐๐๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ + ๐๐๐ง๐ง)
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Hydrostatic Stress
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โข Hydrostatic stress: The average normal stress๐๐โ =
๐๐๐ฅ๐ฅ + ๐๐๐ฆ๐ฆ + ๐๐๐ง๐ง3
Substituting this into volumetric strain for an isotropic material,
๐๐๐ฃ๐ฃ =3 1 โ 2๐๐
๐ธ๐ธ๐๐โ
โข Bulk modulus: The constant of proportionality between volumetric strain and hydrostatic stress
๐ต๐ต =๐๐โ๐๐๐ฃ๐ฃ
=๐ธ๐ธ
3 1 โ 2๐๐
โข ๐๐๐ฃ๐ฃ and ๐๐โ are invariant quantities which always have the same values, regardless of the choice of coordinate system.
โข In other words, the sum of the normal strains and the sum of the normal stresses will have the same value for any coordinate system.
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Thermal Strains
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โข Thermal Strain: Elastic strain caused by the temperature changes๐๐ = ๐ผ๐ผ ๐๐ โ ๐๐0 = ๐ผ๐ผ(โ๐๐)
Figure 5.11 Force vs. distance between atoms. A thermal oscillation of equal potential energies about the equilibrium position xe gives an average distance xavg greater than xe.
Figure 5.12 Coefficients of thermal expansion at room temperature versus melting temperature for various materials. (Data from [Boyer 85] p. 1.44, [Creyke 82] p. 50, and [ASM 88] p. 69.)
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5.4 Anisotropic Materials
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Figure 5.14 Anisotropic materials: (a) metal plate with oriented grain structure due to rolling, (b) wood, (c) glass-fiber cloth in an epoxy matrix, and (d) a single crystal.
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Anisotropic Hookeโs Law
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โข The general anisotropic form of Hookeโs law:๐๐๐ฅ๐ฅ๐๐๐ฆ๐ฆ๐๐๐ง๐ง๐พ๐พ๐ฆ๐ฆ๐ง๐ง๐พ๐พ๐ง๐ง๐ฅ๐ฅ๐พ๐พ๐ฅ๐ฅ๐ฆ๐ฆ
=๐๐11 โฏ ๐๐16โฎ โฑ โฎ๐๐16 โฏ ๐๐66
๐๐๐ฅ๐ฅ๐๐๐ฆ๐ฆ๐๐๐ง๐ง๐๐๐ฆ๐ฆ๐ง๐ง๐๐๐ง๐ง๐ฅ๐ฅ๐๐๐ฅ๐ฅ๐ฆ๐ฆ
โข ๐๐๐๐๐๐ change if the orientation of the x-y-z coordinate system is changed.โข Each unique ๐๐๐๐๐๐ has a different nonzero value.โข The matrix is symmetrical with 21 independent variables.โข In the isotropic case,
๐๐๐๐๐๐ =
1/๐ธ๐ธ โ๐๐/๐ธ๐ธ โ๐๐/๐ธ๐ธโ๐๐/๐ธ๐ธ 1/๐ธ๐ธ โ๐๐/๐ธ๐ธโ๐๐/๐ธ๐ธ โ๐๐/๐ธ๐ธ 1/๐ธ๐ธ
0
01/๐บ๐บ 0 0
0 1/๐บ๐บ 00 0 1/๐บ๐บ
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Orthotropic Materials; Other Special Cases
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โข Orthotropic material: The material that possesses symmetry about three orthogonal planesโ The coefficients for Hookeโs law for an orthotropic material:
๐๐๐๐๐๐ =
1/๐ธ๐ธ๐๐ โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ โ๐๐๐๐๐๐/๐ธ๐ธ๐๐โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ 1/๐ธ๐ธ๐๐ โ๐๐๐๐๐๐/๐ธ๐ธ๐๐โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ 1/๐ธ๐ธ๐๐
0
01/๐บ๐บ๐๐๐๐ 0 0
0 1/๐บ๐บ๐๐๐๐ 00 0 1/๐บ๐บ๐๐๐๐
โข Cubic material: The material that has the same properties in the X-, Y-, and Z-directionsโ Three independent constants: ๐ธ๐ธ๐๐,๐บ๐บ๐๐๐๐, ๐๐๐๐๐๐โ There is still one more independent constant than for the isotropic case, and
the elastic constants still apply only for the special X-Y-Z coordination system.โข Transversely isotropic material: The properties are the same all
directions in a plane, such as the X-Y planeโ Five independent constants: ๐ธ๐ธ๐๐, ๐๐๐๐๐๐,๐ธ๐ธ๐๐, ๐๐๐๐๐๐,๐บ๐บ๐๐๐๐
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Orthotropic Materials
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Orthogonal planes of symmetry
Orthotropic material Transversely isotropic material
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Fibrous Composites
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โข For thin plates or sheets, we assume that ๐๐๐๐ = ๐๐๐๐๐๐ = ๐๐๐๐๐๐ = 0.โข Therefore, Hookeโs law can be used in following form:
๐๐๐๐๐๐๐๐๐พ๐พ๐๐๐๐
=1/๐ธ๐ธ๐๐ โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ 0
โ๐๐๐๐๐๐/๐ธ๐ธ๐๐ 1/๐ธ๐ธ๐๐ 00 0 1/๐บ๐บ๐๐๐๐
๐๐๐๐๐๐๐๐๐๐๐๐๐๐
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Elastic Modulus Parallel to Fibers (1)
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Figure 5.15 Composite materials with various combinations of stress direction and unidirectional reinforcement. In (a) the stress is parallel to fibers, and in (b) to sheets of reinforcement, whereas in (c) and (d) the stresses are normal to similar reinforcement.
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Elastic Modulus Parallel to Fibers (2)
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โข Fibers: isotropic material with elastic constant ๐ธ๐ธ๐๐ , ๐๐๐๐, and ๐บ๐บ๐๐โข Matrix: isotropic material with elastic constant ๐ธ๐ธ๐๐, ๐๐๐๐, and ๐บ๐บ๐๐โข Consider a uniaxial stress ๐๐๐๐ parallel to fibers.โข Assumption: The fibers are perfectly bonded to the matrix.
๐ด๐ด = ๐ด๐ด๐๐ + ๐ด๐ด๐๐๐๐๐๐๐ด๐ด = ๐๐๐๐๐ด๐ด๐๐ + ๐๐๐๐๐ด๐ด๐๐
๐๐๐๐ = ๐ธ๐ธ๐๐๐๐๐๐, ๐๐๐๐ = ๐ธ๐ธ๐๐๐๐๐๐ , ๐๐๐๐ = ๐ธ๐ธ๐๐๐๐๐๐
From the assumption (๐๐๐๐ = ๐๐๐๐ = ๐๐๐๐), yielding the modulus of the composite material:
๐ธ๐ธ๐๐ =๐ธ๐ธ๐๐๐ด๐ด๐๐ + ๐ธ๐ธ๐๐๐ด๐ด๐๐
๐ด๐ดVolume fractions:
๐๐๐๐ =๐ด๐ด๐๐๐ด๐ด , ๐๐๐๐ = 1 โ ๐๐๐๐ =
๐ด๐ด๐๐๐ด๐ด
Thus,๐ธ๐ธ๐๐ = ๐๐๐๐๐ธ๐ธ๐๐ + ๐๐๐๐๐ธ๐ธ๐๐
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Elastic Modulus Transverse to Fibers
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โข Consider uniaxial stress ๐๐๐๐ in orthogonal in-plane direction.๐๐๐๐ = ๐๐๐๐ = ๐๐๐๐
๐๐๐๐ =โ๐๐๐๐
, ๐๐๐๐ =โ๐๐๐๐๐๐๐๐
, ๐๐๐๐ =โ๐๐๐๐๐๐๐๐
whereโ๐๐ = โ๐๐๐๐ + โ๐๐๐๐
Therefore,
๐๐๐๐ =๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐
๐๐1๐ธ๐ธ๐๐
=1๐ธ๐ธ๐๐๐๐๐๐๐๐ +
1๐ธ๐ธ๐๐
๐๐๐๐๐๐
Volume fractions:
๐๐๐๐ =๐๐๐๐๐๐ , ๐๐๐๐ = 1 โ ๐๐๐๐ =
๐๐๐๐๐๐
Thus,1๐ธ๐ธ๐๐
=๐๐๐๐๐ธ๐ธ๐๐
+๐๐๐๐๐ธ๐ธ๐๐
, ๐ธ๐ธ๐๐ =๐ธ๐ธ๐๐๐ธ๐ธ๐๐
๐๐๐๐๐ธ๐ธ๐๐ + ๐๐๐๐๐ธ๐ธ๐๐
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Other Elastic Constants, and Discussion
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โข Estimate of the major Poissonโs ration, ๐๐๐ฟ๐ฟ๐ฟ๐ฟ๐๐๐๐๐๐ = ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐
โข Estimate of the shear modulus
๐บ๐บ๐๐๐๐ =๐บ๐บ๐๐๐บ๐บ๐๐
๐๐๐๐๐บ๐บ๐๐ + ๐๐๐๐๐บ๐บ๐๐
โข Actual values of ๐ธ๐ธ๐๐ are usually reasonably close to the estimate.โข Since ๐ธ๐ธ๐๐ is a lower bound for the case of fibers, actual values are
somewhat higher.โข Quasi-isotropic: A material whose elastic constants are approximately
the same for any direction in the X-Y plane, but different in the Z-direction
โ Isotropic material for in-plane loadingโ Transversely isotropic material for general three-dimensional analysis