I also faced a similar problem with android phone in past.
Here we are explaining some of the common reasons that why internet not working on android phone after connected to Wi Fi. Solution -4 (Important) Update your phone, Modem& Wi-Fi Router time and date to match the internet provider time and date.Before proceeding to the solution you need to verify that the problem is with your android phone & not with your home internet connection or router. This is very important because 2 out of 10 cases are because of this issue.To verify this, please connect you Wi-Fi with other computer/phone. Solution-5 Advance solutions for Android connected to wifi but no internet access Solution-1 Open your router page and rename the default IP from 192.168.1.1 to some other value like 192.168.1.2 or any other number.If you are able to access the internet on other device then you have to check your Android Phone. Reboot the router, click on forget the network on your phone and try to connect again.If you are not able to access the internet on other devices then you have to check you internet connection. Now remember your router default IP is changed to 192.168.1.2 and router page will open by typing 192.168.1.2 in browser window instead of 192.168.1.1.
Solution-2 Open the router page again and try to change the Security Mode under Wireless from default WEP to WPA or if you had selected WAP then choose WPA2. Solution-3 You can check the internet connectivity after changing the encryption from AES to TKIP or vice versa or TKIP & AES.
You have to check the phone internet connectivity on all condition.
You can also call your internet service provider for all these router settings to work with your phone.
Solution-4 In your router you can also check the network mode like B, G, N or BG-mixed & others.
You can also find standard channel also from 1 to 11/12.
You can check the best settings with your Internet service provider. If your using Android version earlier than 4.2 (Jelly Bean) Open Setting Now click on Wireless & settings Now click on Wi-Fi settings & now click on left Menu Key Now click on Advance & select Use Static IP Now you have to contact you Internet Service provider and ask them for IP Address, Gateway, Subnet Mask, DNS1 and DNS2.