### Prior Knowledge to Use
Consider using the following well-known elastic constitutive models as priors or for inspiration:


```python
class LinearElasticity(nn.Module):

    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}):
        """
        Define trainable continuous physical parameters for differentiable optimization.
        Tentatively initialize the parameters with the default values in args.

        Args:
            youngs_modulus_log (float): log of Young's modulus.
            poissons_ratio (float): Poisson's ratio.
        """

        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log)) 
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio)) 

    def forward(self, F: torch.Tensor) -> tuple[torch.Tensor, torch.Tensor]:
        """
        Compute updated Kirchhoff stress tensor from deformation gradient tensor.
        Args:
            F (torch.Tensor): deformation gradient tensor (B, 3, 3).

        Returns:
            kirchhoff_stress (torch.Tensor): Kirchhoff stress tensor (B, 3, 3).
        """

        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = youngs_modulus / (2 * (1 + poissons_ratio))
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        I = torch.eye(3, dtype=F.dtype, device=F.device).unsqueeze(0) # (1, 3, 3)

        Ft = F.transpose(1, 2) # (B, 3, 3)
        F_trace = F.diagonal(dim1=1, dim2=2).sum(dim=1).view(-1, 1, 1) # (B, 1, 1)

        pk_stress = mu * (F + Ft - 2 * I) + la * (F_trace - 3) * I # (B, 3, 3)
        kirchhoff_stress = torch.matmul(pk_stress, Ft) # (B, 3, 3)

        return kirchhoff_stress

class VolumeElasticity(nn.Module):
    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}):
        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: Tensor) -> Tensor:
        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        J  = torch.det(F).view(-1, 1, 1)
        I  = torch.eye(3, dtype=F.dtype, device=F.device).unsqueeze(0)

        kirchhoff_stress = la * J * (J - 1) * I
        return kirchhoff_stress


class SigmaElasticity(nn.Module):
    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}) -> None:
        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: Tensor) -> Tensor:
        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = youngs_modulus / (2 * (1 + poissons_ratio))
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        U, sigma, Vh = torch.linalg.svd(F)
        sigma = torch.clamp_min(sigma, 1e-5)

        epsilon = sigma.log()
        trace = epsilon.sum(dim=1, keepdim=True)
        tau = 2 * mu * epsilon + la * trace

        kirchhoff_stress = U @ torch.diag_embed(tau) @ U.transpose(1, 2)
        return kirchhoff_stress


class CorotatedElasticity(nn.Module):
    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}) -> None:
        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: Tensor) -> Tensor:
        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = youngs_modulus / (2 * (1 + poissons_ratio))
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        U, sigma, Vh = torch.linalg.svd(F)
        sigma = torch.clamp_min(sigma, 1e-5)

        R = U @ Vh
        corotated_stress = 2 * mu * (F - R) @ F.transpose(1, 2)

        J = torch.prod(sigma, dim=1).view(-1, 1, 1)
        I = torch.eye(F.size(-1), dtype=F.dtype, device=F.device).unsqueeze(0)
        volume_stress = la * J * (J - 1) * I

        kirchhoff_stress = corotated_stress + volume_stress
        return kirchhoff_stress


class FluidElasticity(nn.Module):
    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}) -> None:
        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: Tensor) -> Tensor:
        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = 0.0  
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        U, sigma, Vh = torch.linalg.svd(F)
        sigma = torch.clamp_min(sigma, 1e-5)

        R = U @ Vh
        corotated_stress = 2 * mu * (F - R) @ F.transpose(1, 2)   

        J = torch.prod(sigma, dim=1).view(-1, 1, 1)
        I = torch.eye(F.size(-1), dtype=F.dtype, device=F.device).unsqueeze(0)
        volume_stress = la * J * (J - 1) * I

        kirchhoff_stress = corotated_stress + volume_stress
        return kirchhoff_stress


class StVKElasticity(nn.Module):
    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}) -> None:
        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: Tensor) -> Tensor:
        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = youngs_modulus / (2 * (1 + poissons_ratio))
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        U, sigma, Vh = torch.linalg.svd(F)
        sigma = torch.clamp_min(sigma, 1e-5)

        Ft  = F.transpose(1, 2)
        FtF = Ft @ F
        I   = torch.eye(F.size(-1), dtype=F.dtype, device=F.device).unsqueeze(0)
        E_GL = 0.5 * (FtF - I)

        stvk_stress = 2 * mu * F @ E_GL

        J = torch.prod(sigma, dim=1).view(-1, 1, 1)
        volume_stress = la * J * (J - 1) * I

        kirchhoff_stress = stvk_stress + volume_stress
        return kirchhoff_stress

class NeoHookeanElasticity(nn.Module):

    def __init__(self, youngs_modulus_log: float = {youngs_modulus_log}, poissons_ratio: float = {poissons_ratio}):
        """
        Define trainable continuous physical parameters for differentiable optimization.
        Tentatively initialize the parameters with the default values in args.

        Args:
            youngs_modulus_log (float): log of Young's modulus.
            poissons_ratio_sigmoid (float): Poisson's ratio before sigmoid.
        """

        super().__init__()
        self.youngs_modulus_log = nn.Parameter(torch.tensor(youngs_modulus_log))
        self.poissons_ratio = nn.Parameter(torch.tensor(poissons_ratio))

    def forward(self, F: torch.Tensor) -> tuple[torch.Tensor, torch.Tensor]:
        """
        Compute updated Kirchhoff stress tensor from deformation gradient tensor.

        Args:
            F (torch.Tensor): deformation gradient tensor (B, 3, 3).

        Returns:
            kirchhoff_stress (torch.Tensor): Kirchhoff stress tensor (B, 3, 3).
        """

        youngs_modulus = self.youngs_modulus_log.exp()
        poissons_ratio = self.poissons_ratio

        mu = youngs_modulus / (2 * (1 + poissons_ratio))
        la = youngs_modulus * poissons_ratio / ((1 + poissons_ratio) * (1 - 2 * poissons_ratio))

        I = torch.eye(3, dtype=F.dtype, device=F.device).unsqueeze(0) # (1, 3, 3)

        Ft = F.transpose(1, 2) # (B, 3, 3)
        FFt = torch.matmul(F, Ft) # (B, 3, 3)
        J = torch.linalg.det(F).view(-1, 1, 1) # (B, 1, 1)

        kirchhoff_stress = mu * (FFt - I) + la * torch.log(J) * I # (B, 3, 3)

        return kirchhoff_stress
```