1. Introduction
A strain gauge is known to produce resistance change due to the induced strain by the applied external loading, which is often used to measure the force, displacement, vibration, associated with the temperature, humidity, and acceleration [
1,
2,
3]. Strain-type gauge elements are typically made of fiber substrate [
4] or metallic resistance materials. Fiber strain gauge shifts the wavelength of light propagating through a fiber Bragg grating (FBG) applied by an external force, while a resistance strain one produces resistance change due to the induced strain within sensitive grids. In a resistance strain gauge, sensitive grids are usually formed as grids patterns to enhance the sensitivity. In either a resistance strain gauge or a fiber grating one, the strain is transmitted from the surface of the elastomer-sensitive element (ESE) passing through a few polymer films to the sensitive grids. These substrate films often refer to adhesive glue layers to bind the ESE and the resistance strain gauge, polymer substrates to support the sensitive grids. When passing through these substrate films, strain experiences inevitable losses, which result in distortion measurement. To understand and compensate the strain losses, lots of effects have been made to model the strain transfer processing. It is highly desirable to develop a strain transfer model to investigate the relationship between the elastic modulus of the substrate materials and the strain transfer ratio (STR) [
5]. A two-dimensional strain transfer model, which considers the dimensional sizes of adhesive glue layers, explicates the mechanism of strain transfer [
6,
7]. An analysis software can also enable the strain transfer model to analyze the three-dimensional resistance strain gauge [
8].
There are many literatures to create the strain transfer theory for fiber strain gauges. The shear lag theory is an important basis for the study of strain transfer. In 1952, Cox proposed the shear lag theory to analyze the stress transfer in fiber composites. The theory indicated that the displacement difference among the substrate materials determine the stress energy loss [
9]. According to the standard shear lag theory and the strain gradient effect, Toll presented a second-order shear lag theory for elastic aligned short-fiber-reinforced composites [
10]. The loading experiment verified the significance of the shear lag theory for the calculation of the strain transfer coefficient [
11]. We applied this theory to study the resistance strain-type transducer. In contrast to the original formula, the theory has been extended. The parameters of polymer and metal materials, and the geometrical dimensional sizes have been taken into account in the extended theory. This theory helps to understand the shear stress distribution and the axial strain distribution within the sensitive grids and the coating layer.
The STR describes the energy percentage transferred to the sensitive grids from the elastomer-sensitive host material. This property is determined by the structural parameters, including the bonding length, film thickness, and elastic modulus of each layer [
12]. Low elastic modulus of the bonding layer results in measurement of various shear stresses along the middle layer between the fiber core and the host structure. A portion of the host material strain is absorbed by the protective coatings, when the strain transfers from the host material to the fiber core, and hence only a small fraction of structural strain is sensed [
13]. The analytical model to characterize the strain transmission in a strain FBG gauge was corrected to make up for the strain transmission loss by considering the elasticity of substrate materials and FBG stiffness [
14]. The STR is one of the commonly used characteristics to evaluate the transmission loss in a strain-type gauge. The STR of a resistance gauge or a fiber grating gauge is mainly determined by the geometric dimensions and the elastic modulus of each layer that encapsulate the sensitive elements [
15].
We investigated the resistance strain-type transducer system based on the elastic–mechanical shear lag theory, and built up a strain transfer model to calculate the strain transfer characteristics. In our study, a resistance strain-type transducer was modeled to be a four-layer and two-glue (FLTG) model, and the strain transfer characteristics were formulated. The dependences of the strain transfer characteristics on the structural parameters were explored, taking into account of dimensional sizes and the elastic modulus. A norm property of strain transitional zone (STZ) was defined. The STR and the defined term were calculated from the induced formula. The most common contributing factors for the strain transfer loss were discussed on the obtained results. From the view of the strain distribution along the sensitive grids, the qualitative conclusions for the strain transfer characteristics were drawn.
3. Strain Transfer Model Analysis
There are many methods to build strain transfer models for analyzing the strain transfer losses, such as the classical mechanics solution, and finite element simulation. The shear lag theory is one of the frequently used methods in studying the transfer characteristics of composite mechanics. This theory is also accepted to analyze fiber strain sensors [
11,
15]. Here, the shear lag theory was used to investigate the strain transfer properties of the resistance strain-type transducer.
Figure 2a,b show the geometrical dimensions in an FLTG model of a resistance strain-type transducer. The cross-section sizes of a single sensitive grid, the film substrate and the ground adhesive glue layer are
,
, and
, respectively. The sizes of
,
,
, and
are the thicknesses of the cover layer, upper adhesive glue, film substrate layer, and ground adhesive glue, respectively. The thickness of the upper adhesive glue,
satisfies
. The upper adhesive glue, film substrate layer, and cover layer have the identical width and length, which are written as
,
. The sensitive grids are embedded in the upper adhesive glue. We assumed that the shear stress is uniformly distributed in the upper adhesive glue layer, which can be written as
,
.
The following assumptions of the elastic theory were adopted as following [
6]:
Under a static loading condition, the structural layers are elastic material. Only the elastic properties of each layer are considered and the plastic deformations are ignored.
The structural layers are combined together without any relative slipping;
The ESE bears a uniform tensile strain in the axial direction;
The sensitive grids are deformed by the adhesive glues and the film substrate indirectly;
The materials deformations due to the temperature and other humidity are ignored.
Figure 2c shows the elastic–mechanical shear lag theory for the strain transfer characteristics in a resistance strain measuring system. The measured strain is applied along the
x axis. In the
y direction,
,
,
,
, where
,
,
and
are the positions of the lower surface of the sensitive grids, upper adhesive glue, film substrate and ground adhesive glue in the
y-axis, which will be used to simplify Equations (1)–(5).
As shown in
Figure 2c, taking into account of a finite element of
in the polymer cover, sensitive grids, upper adhesive glue, film substrate and ground adhesive glue, from equilibrium along the
x direction of the sensitive grids, the shear stress can be obtained as follows:
For a polymer cover layer, its resultant force is zero along the
x direction, as shown in
Figure 2c. Then, the shear stress of the polymer cover layer can be further expressed as:
Figure 2 shows the direction of the shear stress. By considering the force equilibrium for an element of the upper adhesive glue in the
x direction, the shear stress in the upper adhesive glue can be expressed as:
Based on the same principle, the equilibrium equations of the film substrate and ground adhesive glue can be obtained:
where
n is the total number of sensitive grids;
is the shear stress in the surface of sensitive grids;
,
, and
refer to the shear stress within the upper adhesive glue, the film substrate layer, and the ground adhesive layer, respectively.
Substituting Equations (1) and (2) into Equation (3), the shear stress inside the upper adhesive glue can be described as:
Following Hook’s law (
), Equations (4)–(6) are rewritten in details as following:
where
and
are the elastic modulus and line strain of composed layers, respectively; the subscripts (c, u, g, m and a) refer to the cover, upper adhesive glue, sensitive grid, film substrate layer, and ground adhesive layer.
The deformations are synchronized within the upper adhesive glue, sensitive grid, film substrate layer and ground adhesive layer. The strain gradients of the upper adhesive glue, sensitive grid, film substrate layer and ground adhesive layer are , , and , respectively.
Since the strain of the upper adhesive glue, sensitive grid, film substrate layer and ground adhesive layer are synchronized, their strain gradients are close to each other, and can be written as:
Because the elastic modulus of the sensitive grid deviates far from those of the other structural layers, the deformation gradients satisfy the following approximate formulae:
Substituting Equations (11)–(13) into Equations (7)–(9), the simplified strains expressions are:
Following Hook’s law (
), the strain can be described as:
The relative displacement between any two relevant layers is integrated from the shear strain
where
is the shear modulus of a material,
is the shear strain, and
is the relative displacement.
Substituting Equations (14)–(16) into Equation (18), the relative displacement between the lower surface of the sensitive grids and the upper adhesive glue, the relative displacement between the upper adhesive glue and the film substrate, and the relative displacement between the film substrate and the ground adhesive glue can be written as following:
Combining Equations (19)–(21), the relative displacement between the lower surface of the sensitive grids and the ground adhesive glue can be described as below:
Equation (22) can be simplified as:
where
Equation (24) describes the shearing-stress transfer parameter, k, which depends on the elastic modulus, shear modulus, and the geometric dimensions of the structural layers.
Taking the derivative of Equation (23) with respect to
x, the differential equation of the relative displacement between the lower surface of the sensitive grids and the ground adhesive glue can be described:
The general solution to Equation (25) is assumed to be
where,
, and
are the shearing-strain of the sensitive grids and the ESE. The two unknowns
and
are solved based on the boundary conditions. The boundary condition is estimated at the condition of
, where the axial load of the sensitive grids is zero, written as:
It should be noted that
. The axial strain distribution within the sensitive grids is
Equation (28) shows that the axial strain distribution is dominated by the shearing-stress transfer parameter, k. Equation (24) shows that k is determined by the size of the structural layers, the elastic modulus of the sensitive grids, the shear modulus of the film substrate, and the ground adhesive layer.
5. Conclusions
The strain transfer characteristics of the resistance strain-type transducer were investigated using the elastic–mechanical shear lag theory. The FLTG model was analyzed in elastic–mechanical shear lag theory, and a strain transfer progress was formulated in theory. The characteristics of the STZ and the STR were obtained by the induced formulae. The dependence of the STZ and its ratio on the structural parameters of the resistance strain-type transducer was calculated. The conclusions obtained in this study confirmed the role of structural parameters in the STZ to ensure reliability of the resistance strain-type transducer system.
Here, we focused the dependence of the adhesive glue on the strain transfer characteristics. The five parameters of the resistance strain-type transducer (including adhesive thickness, adhesive width, shear modulus and grids length) were analyzed. Specific conclusions are as follows:
(1) The increasing of the lateral width of the ground adhesive glue layer and the shear modulus, the reduction of the thickness of above ground adhesive glue layer, will reduce the STR at both ends of the resistance strain grid. The bonding process of the resistance strain-type transducer into the surface of ESE must be strictly controlled to ensure adequate bonding strength and insulation resistance. At the same time, it required that ground adhesive glue had twice the area of the film substrate layer. The ground adhesive glue layer should be as thin as possible, while its shear modulus should be as large as possible. In order to ensure a higher sensitivity of the resistance strain-type transducer, it is necessary that the STZ ratio was less than 10%.
(2) The increasing of the sensitive grid length could improve the STR, whereas the increasing of the sensitive grid width will decrease the STR. The end-effect due to the grid width was reduced by increasing the grids length or by optimizing the grids patterns. Selecting a ground adhesive glue with a large elasticity modulus could effectively reduce the influence of the thickness of the ground adhesive glue on the STR.
However, the effect of temperature changes on the elastic modulus of the adhesive glue was not negligible [
23], and the effect of temperature on the performance of the resistance strain-type transducer will be investigated in future work.