### Test results

As shown in Fig. 4a, all the specimens are damaged at the effective part, that is, the part with the smallest cross section of the sample. Unlike the aluminum test results, the failure section of the A-WPC is almost parallel to the horizontal axis of the sample, and no obvious necking is observed, as shown in Fig. 4b. This indicates that the WPC can effectively prevent the aluminum from necking when bearing a uniaxial tensile load. After the failure of the specimens, they all bent towards the WPC and formed into arcs, as shown in Fig. 4c. Different elastic modulus lead to the unequal residual strains of the two materials after fracture springback, resulting in bending deformation of the whole sample.

The experimental results of the A-WPC are shown in Table 2. The coefficient of variation of all the parameters is less than 17%, meeting the specification requirements. Meanwhile, only the coefficient of variation of the elongation at break is more than 10%, which is 12%. The *R*^{2} of the ultimate strength and elastic modulus are less than 5%, which shows that the A-WPC continues to exhibit the advantages of good homogeneity, stable mechanical properties, and low dispersion of the WPC.

Figure 5 shows the tensile stress–strain relationship of the A-WPC, which can be divided into three stages: elastic stage, elastic–plastic stage, and ductile failure stage. At first, the relationship between the stress and strain is linear. After reaching the proportional limit, the slope of the curve becomes smaller, and the curve enters the elastic–plastic stage. In this stage, the growth rate of stress decreases relative to the growth rate of strain, and finally, it enters the ductile failure stage. At this time, the strain increases continuously, the stress increases slowly, and the sample undergoes a large deformation, which displays brilliant ductility until the sample fails.

### Calculation model

#### Equivalent elastic modulus

The Poisson’s ratio of the WPC and aluminum is consistent, which possesses the basic elements of deformation coordination, and no relative slip appears in the tensile process of the A-WPC, indicating that the A-WPC has good mechanical stability. Thus, the equivalent stiffness method can be used to calculate the elastic modulus of the A-WPC, as shown in the following equation:

$$ E = \frac{{E_{{\text{W}}} wt_{{\text{w}}} + E_{{\text{A}}} wt_{{\text{A}}} }}{{wt_{{\text{w}}} + wt_{{\text{A}}} }} = \frac{{E_{{\text{W}}} t_{{\text{w}}} + E_{{\text{A}}} t_{{\text{A}}} }}{{t_{w} + t_{{\text{A}}} }}, $$

(1)

where *E*_{W} (*E*_{A}) and *t*_{W} (*t*_{A}) are the elastic modulus and the thickness of the WPC (aluminum), respectively, and *w* is the width of the sample. By substituting the elastic modulus of the WPC and aluminum into Eq. (1), the calculated value of the elastic modulus of the A-WPC is 27.66 GPa, which is different from the experimental result by only 12.8%.

#### Model I—bilinear model

The elastic–plastic stage, as the transition section between the elastic and the ductile failure stage, can be decomposed into the extended section of the elastic stage and the initial stage of the ductile failure section. Then, the stress–strain relationship can be disassembled into two linear curves. The slope of the first straight line is the elastic modulus, and the slope of the curve drops sharply after reaching the inflection point strength; therefore, the bilinear model including two straight lines can be written as the following equation:

$$ f = \left\{ {\begin{array}{*{20}l} {E\varepsilon } & {0 < \varepsilon \le \varepsilon_{{{\text{ey}}}} } \\ {\sigma_{{{\text{ey}}}} + kE\left( {\varepsilon - \varepsilon_{{{\text{ey}}}} } \right)} & {\varepsilon_{{{\text{ey}}}} < \varepsilon \le \varepsilon_{{\text{u}}} } \\ \end{array} } \right., $$

(2)

where *σ*_{ey} and *ε*_{ey} are the inflection point strength and homologous strain of the A-WPC, respectively, which is between the proportional limit and yield strength point, and *k* is a constant. According to the tensile test results, by setting the inflection point strength to be 90% of the ultimate strength, the inflection point strain is the strain corresponding to the inflection point strength, and *k* is suggested to be 0.03. Substituting the above parameters into Eq. (2):

$$ f = \left\{ {\begin{array}{*{20}l} {24510\varepsilon } & {0 < \varepsilon \le 0.0031} \\ {75.98 + 735\left( {\varepsilon - 0.0031} \right)} & {0.0031 < \varepsilon \le 0.017} \\ \end{array} } \right.. $$

(3)

Figure 6 shows the fitting results of the A-WPC tensile stress–strain bilinear model. The model is simple in form and easy for hand computation. The two segments can well describe the stress–strain relationship at different stages and can also predict the elastic modulus, ultimate strength, and ultimate strain of the materials accurately.

#### Model II—exponential model

In the light of the tensile test characteristics of the A-WPC, the exponential function model (Eq. 4) is introduced to predict the tensile stress–strain relationship:

$$ f\left( x \right) = \frac{\sigma }{{\sigma_{{\text{y}}} }} = e^{bx} - e^{dx} , $$

(4)

where *x* = *ε*/*ε*_{y}, *σ*_{y} and *ε*_{y} represent the yield stress and yield strain, respectively, and *b* and *d* are constants that can be obtained from the test results. Equation (4) is derived, and *x* = 0 is taken into Eq. (5), where *E* is the elastic modulus and *E*_{sec} is the secant modulus of the yield point:

$$ b - d = \frac{E}{{E_{{{\text{sec}}}} }}. $$

(5)

Substituting the test results into Eq. (5) for the calculation and setting *σ*_{y} to be 95% of the ultimate strength, *ε*_{y} is the strain corresponding to the yield point. After regression analysis, we suggest that *b* = 0.02 and *d* = − 1.55. Taking these parameters into Eq. (4), the expression of the tensile stress–strain relationship of the A-WPC can be obtained as

$$ \sigma = 79.91\left( {e^{6.06\varepsilon } - e^{ - 469.7\varepsilon } } \right). $$

(6)

Figure 7 shows that the trend of the stress–strain relationship between the exponential model and the test results of the A-WPC is almost the same, which means this model can describe the whole process of the stress–strain relationship development. When the numerical simulation and software analysis of the A-WPC are carried out, the exponential model can get more accurate results.

### Analysis of tensile springback of the A-WPC

Throughout the experiment, no interface relative slip and interface delamination occurred in all specimens, which indicates that the WPC and aluminum are in a state of coordinated deformation in the whole process of the trial. The eccentric tensile load and additional bending moment can be ignored, and the tensile process can be regarded as a plane stress state. As shown in Fig. 8, the tensile springback process of the A-WPC is as follows: assuming that the original length of the sample is *L*, the length after the uniaxial tensile load is *L* + 2*δ*_{u}, and the tensile strain of the sample is *ε*_{u}, when unloading, the sample begins to shrink and spring back, and bending occurs. The springback is divided into the springback shrinkage stage and the springback bending stage. Supposing that the sample only shrinks and does not bend in the springback shrinkage stage, the sample will rebound until the resultant stress of the section is 0. Then, in the springback bending stage, the sample will bend under the action of the pure bending moment, the stress of the section will redistribute, and the final bending moment of the section will be 0.

Let us assume that the constitutive relationship of the aluminum and WPC can be described by the bilinear model, which is divided into the elastic section and linear strengthening section. In this model, *E*_{a}, *E*_{a}′, *σ*_{ay}, *ε*_{ay}, and *t*_{a} are the elastic value of the aluminum elastic section, the elastic modulus of the linear strengthening section, the yield strength, the yield strain, and the thickness, respectively. The corresponding parameters of the WPC are *E*_{w}, *E*_{w}′, *σ*_{wy}, *ε*_{wy}, and *t*_{w}, where *ε*_{ay} = *σ*_{ay}/*E*_{a} and *ε*_{wy} = *σ*_{wy}/*E*_{w}. The thickness of the A-WPC is *t* = *t*_{a} + *t*_{w}, the actual thickness coordinate is *h*, the section residual strain is *ε*_{c}(*h*), and the bending residual curvature is *A* = *ε*_{c}(*h*)/*h*.

When the longitudinal section with a residual strain of 0 is defined as the neutral axis of the residual strain, the offset of the neutral axis from the interface is *δ*. When the tensile strain is defined as *ε*_{k}, the stress of the aluminum is *σ*_{ak} and the stress of the WPC is *σ*_{wk}; when the rebound strain is defined as *ε*_{t}, the stress of the aluminum is *σ*_{at} and the stress of the WPC is *σ*_{wt}; the residual stress is *ε*_{c} (*z*).

When the tensile strain is *ε*_{k}, the stresses of the WPC and aluminum could be expressed by Eqs. (7) and (8):

$$ \sigma_{{{\text{wk}}}} = \left\{ {\begin{array}{*{20}l} {E_{{\text{w}}} \varepsilon_{{\text{k }}} } & {\varepsilon_{{\text{k }}} \le \varepsilon_{{{\text{wy}}}} } \\ {E_{{\text{w}}} \varepsilon_{{{\text{wy}}}} + E_{{\text{w}}}^{^{\prime}} \left( {\varepsilon_{{\text{k }}} - \varepsilon_{{{\text{wy}}}} } \right)} & {\varepsilon_{{\text{k }}} > \varepsilon_{{{\text{wy}}}} } \\ \end{array} } \right.. $$

(7)

$$ \sigma_{{{\text{ak}}}} = \left\{ {\begin{array}{*{20}l} {E_{{\text{a}}} \varepsilon_{{\text{k }}} } & {\varepsilon_{{\text{k }}} \le \varepsilon_{{{\text{ay}}}} } \\ {E_{{\text{a}}} \varepsilon_{{{\text{ay}}}} + E_{{\text{a}}}^{^{\prime}} \left( {\varepsilon_{{\text{k }}} - \varepsilon_{{{\text{ay}}}} } \right)} & {\varepsilon_{{\text{k }}} > \varepsilon_{{{\text{ay}}}} } \\ \end{array} } \right.. $$

(8)

When the springback strain is *ε*_{t}, the stresses of the aluminum and WPC follows:

$$ \left\{ {\begin{array}{*{20}l} {\sigma_{{{\text{at}}}} = \sigma_{{{\text{ak}}}} - E_{{\text{a}}} \varepsilon_{{\text{t}}} } \\ {\sigma_{{{\text{wt}}}} = \sigma_{{{\text{wk}}}} - E_{{\text{w}}} \varepsilon_{{\text{t}}} } \\ \end{array} } \right.. $$

(9)

There is no bending of the A-WPC during the springback shrinkage stage. When the resultant force of the internal force of the section is 0, there is the following relationship: *t*_{a}*σ*_{at} + *t*_{w}*σ*_{wt} = 0. Taking Eq. (9) into it leads to the following equation:

$$ \varepsilon_{{\text{t}}} = \frac{{t_{{\text{a}}} \sigma_{{{\text{ak}}}} + t_{{\text{w}}} \sigma_{{{\text{wk}}}} }}{{t_{{\text{a}}} E_{{\text{a}}} + t_{{\text{w}}} E_{{\text{w}}} }}. $$

(10)

The internal moment *M*_{t} can be expressed as Eq. (11):

$$ M_{{\text{t}}} = \frac{{t_{{\text{a}}}^{2} }}{2}\left( {\sigma_{{{\text{ak}}}} - E_{{\text{a}}} \varepsilon_{{\text{t}}} } \right) + \frac{{t_{{\text{w}}}^{2} }}{2}\left( {\sigma_{{{\text{wk}}}} - E_{{\text{w}}} \varepsilon_{{\text{t}}} } \right). $$

(11)

Under a pure bending load, the distance between the strain neutral axis and the interface is *δ*, and the rotation angle of the section relative to the *z*-axis is *θ* (*δ* and *θ* can be positive or negative). Under this condition, the bending normal stress of the aluminum is *σ*_{ab} and that of the WPC is *σ*_{wb}. The strain distribution of the pure bending section of the A-WPC is shown in Fig. 9.

Supposing that *θ* is small enough, now the strain distribution function along the *z*-axis is

$$ \varepsilon (z) = {\text{ tan}}\theta \left( {z - \delta } \right). $$

(12)

The strain is replaced by tan *θ* (*z* − *δ*), and the bending normal stress could be expressed as

$$ \left\{ {\begin{array}{*{20}l} {\sigma_{{{\text{wB}}}} = E_{{\text{w}}} \tan \theta \left( {z - \delta } \right)} & {0 < z \le t_{{\text{w}}} } \\ {\sigma_{{{\text{aB}}}} = E_{{\text{a}}} \tan \theta \left( {z - \delta } \right)} & { - t_{{\text{a}}} < z \le 0} \\ \end{array} } \right.. $$

(13)

The resultant force of the pure bending normal stress distribution along the section is 0, that is, the sum of the bending normal stress of the A-WPC along the *z*-axis is 0, as shown in the following equation:

$$ \mathop \smallint \limits_{0}^{{t_{{\text{w}}} }} \sigma_{{{\text{wB}}}} {\text{d}}z + \mathop \smallint \limits_{{ - t_{{\text{a}}} }}^{0} \sigma_{{{\text{aB}}}} {\text{d}}z = 0. $$

(14)

Taking Eqs. (13) into (14), *δ* can be determined by the following equation:

$$ \delta = \frac{{E_{{\text{w}}} t_{{\text{w}}}^{2} + E_{{\text{a}}} t_{{\text{a}}}^{2} }}{{2\left( {E_{{\text{w}}} t_{{\text{w}}} - E_{{\text{a}}} t_{{\text{a}}} } \right)}}. $$

(15)

When the bending moment produced by the bending normal stress on the section is *M*, then Eq. (14) becomes

$$ \mathop \smallint \limits_{0}^{{t_{{\text{w}}} }} \sigma_{{{\text{wB}}}} {\text{d}}z + \mathop \smallint \limits_{{ - t_{{\text{a}}} }}^{0} \sigma_{{{\text{aB}}}} {\text{d}}z = M. $$

(16)

Then, taking Eq. (13) into Eq. (16), we can figure out the expression of tan*θ:*

$$ \tan \theta = \frac{M}{{E_{{\text{w}}} \left(\frac{1}{3}t_{{\text{w}}}^{3} - \frac{1}{2}\delta t_{{\text{w}}}^{2} \right)- E_{{\text{a}}} \left( {\frac{1}{3}t_{{\text{a}}}^{3} + \frac{1}{2}\delta t_{{\text{a}}}^{2} } \right)}} $$

(17)

If *M* = − *M*_{t}, the transformation process of the section stress is considered to be the process of the bending moment *M*_{t} formed in the springback shrinkage stage and further releasing of the bending moment, i.e., the rebound bending stage. The superposition of the two stress states determined by Eqs. (9) and (13) is the residual stress state of the section after tensile springback. The residual curvature of the A-WPC can be regarded as the curvature caused by the bending moment *M*_{t}. According to the definition of the residual curvature and Eq. (12), *A* ≈ tan*θ*/2 can be calculated. By substituting Eqs. (5) and (11) into *A* ≈ tan*θ*/2, the final residual curvature of the A-WPC can be expressed as

$$ A = \frac{{t_{{\text{a}}}^{2} \left( {\sigma_{{{\text{ak}}}} - E_{{\text{a}}} \varepsilon_{{\text{t}}} } \right) + t_{{\text{w}}}^{2} \left( {\sigma_{{{\text{wk}}}} - E_{{\text{w}}} \varepsilon_{{\text{t}}} } \right)}}{{4\left[ {E_{{\text{a}}} \left( {\frac{{t_{{\text{a}}}^{3} }}{3} + \frac{\delta }{2}t_{{\text{a}}}^{2} } \right) - E_{{\text{w}}} \left( {\frac{{t_{{\text{w}}}^{3} }}{3} - \frac{\delta }{2}t_{{\text{w}}}^{2} } \right)} \right]}}. $$

(18)

Substituting the test results into Eqs. (10) and (15), the springback strain and the offset of the neutral axis relative to the interface are calculated as *ε*_{t} = 0.0033 and *δ* = − 1.13 mm, respectively. Then, taking the above two results into Eq. (18), *A* = 0.28/m^{−1} can be calculated. The residual strain difference of the test results of the samples labelled as 1–1–1–6 is brought into the formula *A* = *ε*_{c}(*h*)/*h*. The results are illustrated in Fig. 10, indicating that the average residual curvature of the tensile springback of the WPC aluminum composite plate is 0.31/m^{−1}, which is 9.7% higher than the theoretical value. This means that Eq. (18) can accurately predict the tensile springback curvature of the A-WPC.