4.3.1Ideal-elastic deformation behavior
4.1.2.1.1a) Elasticity law
Formally, ideal-elastic deformation behavior is described by the elasticity law:
Equation 4.10
τ = G · γ
Robert Hooke (1635 to 1703) wrote in 1676, in his textbook “de potentia restitutiva” [4.13]: Deformation of solids is proportional to the applied force (“ut tensio sic vis”; meaning: as the extension so is the force). Based on later works on solid-state physics and mechanics by Jakob Bernoulli (in 1689) and Leonhard Euler (in 1736/1765, “mechanica” [4.14]), finally Augustin L. Cauchy (in 1827 [4.15]) stated the modern form of what was called later Hooke’s elasticity law.
If the τ/γ-function is presented in a diagram, ideal-elastic behavior is displayed in the form of a straight line coming from the origin point, showing a constant slope. This slope value corresponds to the value of G. When presented as a function of G(γ) or G(τ), both curves indicate a constant plateau value for G if the limiting value of the linear-elastic range is not exceeded.
The values of the shear modulus of ideal-elastic or Hookean solids are independent of the degree and duration of the shear load applied, when remaining within the linear-elastic range.
For tensile tests, similarly to shear tests, the elasticity law applies in the following form (see also Chapter 4.2.2):
Equation 4.11
σ = E · ε
4.1.2.1.2b) The spring model
The spring model is used to illustrate the behavior of ideal-elastic or Hookean solids (see Figure 4.2).
Figure 4.2: The spring model to illustrate
ideal-elastic behavior
4.1.2.1.3Ideal-elastic behavior, explained by the behavior of a spring
4.1.2.1.41) When under load
Under a constant force, the spring shows immediately step-like deformation, remaining at a constant value as long as the force is applied. When applying forces of differing strength to the spring, it can be observed in all cases: The resulting deflection is proportional to the force applied. The proportionality factor corresponds to the rigidity of the spring, i. e., to the spring constant.
4.1.2.1.52) When removing the load
As soon as the force is removed, the spring recoils elastically, and this means immediately, step-like and completely, returning to the initial state. No deformation remains finally at all.
Comparison: This is in contrast to a Newtonian liquid whose behavior can be illustrated using the dashpot model (see Chapter 2.3.1b).
Summary: Behavior of the spring model
Under a constant load, the spring deforms immediately and remains deformed as long as the load is applied. After removing the load, the previously occurring deformation disappears immediately and completely. In other words: After a load-and-removal cycle, an ideal-elastic material completely returns to the initial state.
Comparison: Metal spring and elasticity law
For tension and compression springs, the force/deflection law or elasticity law according to Hooke holds:
Equation 4.12
F = CH · s
with the spring force F [N], the spring constant CH [N/m] which is the rigidity of the spring and the index H is due to Hooke; and the deflection s [m] of the spring.
Here: F corresponds to the shear stress τ, CH corresponds to the shear modulus G, and s corresponds to the shear deformation γ.
Note: Elastic behavior , and stored deformation energy
Deformation energy acting on an ideal-elastic body during a shear process will be completely stored within the deformed material. When the load is removed, the stored energy can be recovered without any loss, enabling the complete reformation of the material. Therefore here, after deformation and reformation, a completely reversible process has taken place since the shape of the sample is unchanged after the experiment is finished.
For all materials showing ideal-elastic deformation behavior, there are interactive forces between their atoms or molecules. As examples for those very dense, stiff and rigid materials can be imagined stone and steel with crystalline structures at room temperature. If the linear-elastic range is exceeded, they show brittle fracture without any sign of time-dependent creep or creep recovery, e. g. in the form of very slow time-dependent deformation, partial reformation or stress relaxation, respectively. These kinds of materials do not show any visco-elastic behavior, since there is absolutely no viscous component available.
4.4Yield point determination using the shear stress/deformation diagram
4.2.1.1.1a) Yield point at the limit of the linear-elastic range, using a single fitting line
Preset is a controlled shear stress function (similar to Figure 3.1 or 3.2). The steps or the ramp, respectively, should begin at shear stress values at least one decade below the assumed yield point and end up at least one decade above that point [4.16]. The yield point τ1 is the shear stress value at which the linear-elastic deformation range is exceeded (Figure 4.3). The measuring points are usually presented on a logarithmic scale, with the shear stress τ [Pa] on the x-axis and the shear deformation γ [%] on the y-axis, as a logarithmic stress/deformation or τ/γ diagram.
The measuring curve shows a constantly rising slope in the range of low values of τ and γ. For analysis, a straight line is fitted on this curve interval, based on the following consideration: In this first interval, the sample shows linear-elastic deformation behavior. Sometimes, this line is also called a “tangent”, and correspondingly, the procedure is termed the “tangent method”. When properly speaking however, tangents are usually adapted to curves and not to points of straight lines. The elasticity law applies here since the increasing values of τ and γ are still proportional in this deformation range. These kinds of samples behave like homogeneously deformable, gel-like or soft solids.
Figure 4.3: Determination of the yield point τ1
at the limit of the linear-elastic deformation range, using a single straight fitting line in the logarithmic tau-gamma diagram
Figure 4.4: Determination of the yield point τ2 using the “tangent crossover point method” in the logarithmic tau-gamma diagram
Summary
The yield point is the one shear stress value at which the range of the reversible elastic deformation